Limit operators techniques on general metric measure spaces of bounded geometry
Raffael Hagger, Christian Seifert
TL;DR
This paper develops a unified framework for limit operator techniques on metric measure spaces of bounded geometry, providing new insights and broadening applications in spectral analysis of band-dominated operators.
Contribution
It introduces core assumptions for defining limit operators and characterizes compactness and Fredholmness in an abstract setting, unifying and extending classical results.
Findings
Unified approach to limit operators on metric measure spaces
Characterization of compactness and Fredholmness via limit operators
Spectral consequences derived from the framework
Abstract
We study band-dominated operators on (subspaces of) -spaces over metric measure spaces of bounded geometry satisfying an additional property. We single out core assumptions to obtain, in an abstract setting, definitions of limit operators, characterizations of compactness and Fredholmness using limit operators; and thus also spectral consequences. In this way, we recover and unify the classical and recent results on limit operator techniques, but also gain new insights and are able to treat further applications.
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Limit operators techniques on general metric measure spaces of bounded geometry
Raffael Hagger111R.H. has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 844451. and Christian Seifert
Abstract
We study band-dominated operators on (subspaces of) -spaces over metric measure spaces of bounded geometry satisfying an additional property. We single out core assumptions to obtain, in an abstract setting, definitions of limit operators, characterizations of compactness and Fredholmness using limit operators; and thus also spectral consequences. In this way, we recover and unify the classical and recent results on limit operator techniques, but also gain new insights and are able to treat further applications.
MSC2010: Primary: 47A53; Secondary: 47B07, 47B35-38, 47L10
Key words: metric measure spaces, bounded geometry, limit operators, Fredholm, essential spectrum
1 Introduction
One method to study Fredholm properties of operators are so-called limit operators. This theory has its roots in [5, 13, 21, 22, 23, 27, 28, 29, 30, 32, 33], see also the monographs [25, 34] for a thorough treatment. In a nutshell, it deals with the following. Given an operator on some -space over a metric measure space , limit operators can be thought of limits of shifted copies of as the shifts tend to the boundary of (or to infinity with respect to the metric ). One then aims to obtain information on by studying its limit operators . One typical statement in this direction is:
- (I)
is compact if and only if all limit operators of are trivial.
Since an operator is Fredholm if and only if it is invertible modulo compact operators, another statement to expect is:
- (II)
is Fredholm if and only if all limit operators are invertible.
Since the essential spectrum of an operator is related to Fredholmness, as a consequence one then obtains:
- (III)
The essential spectrum of coincides with the union of the spectra of its limit operators.
Classical results in this direction dealt with operators on scalar-valued -spaces over or and its generalizations to for the reflexive range [21]. To treat the the endpoint cases p\in\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}1,\infty\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} and vector-valued analogs one needs a further ingredient, the so-called -theory [25, 34, 36]. Although the method of limit operators can be considered to be classical, surprisingly, significant progress towards statement (II) was made only recently. Even more surprisingly, the result was first shown for p\in\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}1,\infty\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} (see [8, 24, 25, 32]) and then generalized to the Wiener algebra for all by an interpolation argument [34]. The much more difficult case of band-dominated operators was then solved by Lindner and Seidel [26] in 2014. Before these recent developments, even for , there was a somewhat “nasty” uniform invertibility condition involved, which was difficult to deal with in applications.
In the last few years, limit operator techniques have been applied to a variety of situations such as operators on Fock spaces [14], on Bergman spaces [18, 19], and also on uniformly discrete metric measure spaces of bounded geometry [38, 42]. Further applications to spectral theory for operator families can be found in [2, 3]. Although they all share the common method it appears that the corresponding techniques are tailor-made for the particular situation. This is the starting point for our paper. The aim of this paper is to single out the assumptions needed to obtain results of type (I), (II), and (III) in the abstract framework of -spaces for metric measure spaces and . In this way, we unify the theory for the different applications in the literature as well as gain new insights into the method. Specifically, we show that there is an interplay between geometric properties of the metric measure space , compactness of subsets of and compact operators, abstract shifts on and on and compactifications of .
We also want to mention that for the Hilbert space case (i.e. ) there are -algebras techniques available to treat (I), (II) and (III) which are not used in this paper, see e.g. [6, 7, 15, 16, 35]. They lead to similar connections with coarse geometry as the methods presented here.
The paper is organised as follows. In Section 2 we review proper metric spaces of bounded geometry and introduce property A*′, which will be needed in the sequel. The name of our property A′* is of course inspired by Yu’s property A [40], which implies property A*′; in fact they are equivalent for our metric spaces. also satisfies property A′* if it has the much more convenient finite asymptotic dimension. Band-dominated operators are then introduced in Section 3. We characterize the space of band-dominated operators and show some algebraic properties of it. Further, we introduce Toeplitz operators in our setting, which is a source for many applications. The core of this paper is Section 4, where we state the assumptions for which we show that the limit operator method yields a rich theory. Here we only work with the Stone-Čech compactification of and its boundary . After introducing the limit operators we focus on compact operators and characterize them as in (I); cf. Corollary 4.24. Then we aim at proving (II) in Theorem 4.38 and, as a consequence, (III) in Corollary 4.39. In order to do this, we need the notion of lower norms of an operator and its properties. Section 5 is devoted to the influence of the compactification of we use. Since we worked with the “largest one” in the previous section, this is now an easy consequence of the universal property of the Stone-Čech compactification. The last Section 6 collects various applications. Here, we recover many statements on limit operators in various settings by just adjusting our abstract framework. Moreover, we present further examples not yet treated in the literature so far.
Although we only consider scalar-valued -spaces and the reflexive range , this is mainly for convenience and readability. We expect to be able to treat the cases p\in\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}1,\infty\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} as well as Banach space-valued Bochner-Lebesgue spaces (where is a Banach space) using the -theory. This will be investigated in a future project. However, the vector-valued case (i.e. if is finite-dimensional) can already be treated with the methods presented here using a simple tensoring trick, which is sketched in Section 6.
We end this introduction with some notation used in the remaining part. For Banach spaces and we write for the bounded linear operators and . The ideal of compact operators is denoted by and , respectively. For a measurable subset of a measure space we write for the indicator function of (which is exactly one on and zero elsewhere). Operators of multiplication by measurable functions are denoted by . In a metric space , we denote by the open ball around of radius . Correspondingly, denotes the closed ball. We write {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}\cdot\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}} for the cardinality function of sets. Norms of vectors and operators will be denoted by and supplemented with an appropriate subscript if necessary. The complement of a set will be denoted by .
2 On bounded geometry and Property A
Let be a metric space with metric . is called of bounded geometry if there exists such that for all there exists such that for all the ball can be covered by at most open balls of radius . In other words, is of bounded geometry if every open ball can be covered by a finite number of balls of radius and this finite number only depends on and not on .
is called proper provided that closed balls are compact. Proper metric spaces are locally compact (hence the Riesz-Markov-Kakutani Theorem applies). Proper metric spaces are also complete, -compact and hence separable.
Lemma 2.1**.**
Let be a separable metric space of bounded geometry (with some ) and . Then there exists and a sequence of Borel subsets of such that
- (a)
* for , ,* 2. (b)
, 3. (c)
* for all ,* 4. (d)
there is a positive integer such that for all the set J_{k}(r):=\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}j\in J:\mathrm{dist}(Q_{j},Q_{k})\leqslant r\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} has at most elements, 5. (e)
for all and the set \mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}j\in J:Q_{j}\cap B(x,s)\neq\varnothing\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} is finite.
Here, denotes the diameter of a set, i.e. , and denotes the distance between two sets, i.e. . We will use the abbreviation \mathrm{dist}(x,Q):=\mathrm{dist}(\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}x\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}},Q) for the distance between a point and a set .
Proof.
For there exists and such that is a maximal -separated sequence in , i.e. for all , , and for all there exists such that . Without loss of generality let J=\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}1,\ldots,{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}J\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} in case is finite and in case is infinite.
Define for . Then for all . Now define and
[TABLE]
for . Then the sequence satisfies (a), (b) and (c). Moreover, we have for all . Therefore, for (d) it suffices to show that there is a constant such that {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}j\in J:x_{j}\in B(x_{k},6r)\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}}\leqslant N for all .
By the bounded geometry assumption there is an and an integer such that every ball can be covered by at most balls of radius , i.e. for every there exist such that . Now as for all with and , every can only contain at most one . Hence {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}j\in J:x_{j}\in B(x_{k},6r)\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}}\leqslant N for all .
To show the last property, let such that . Since we observe . Hence, if also such that , then . We now reason as above. Let and such that . Since and for all l\in\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}1,\ldots,N\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}, we conclude that every can contain at most one . Hence {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}k\in J:x_{k}\in B(x_{j},4r+2s)\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}}\leqslant N, and therefore {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}j\in J:Q_{j}\cap B(x,s)\neq\varnothing\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}}\leqslant N. ∎
Let be a Banach space, a map and . We say that has -variation if for all satisfying we have {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}\xi(x)-\xi(y)\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\leqslant\varepsilon.
Definition 2.2** ([39, Definition 5.2.2]).**
Let be a proper metric space of bounded geometry. Then has property A provided for all there exists a weak∗-continuous map (the regular countably additive Borel measures on ) such that
- (a)
{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}\mu_{x}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}=1 for all , 2. (b)
has -variation, 3. (c)
there exists such that for all the functional (=complex Radon measure) is supported in .
Definition 2.3**.**
Let be a proper metric space of bounded geometry. We say that has property A′ if for every there is a countably infinite collection of non-zero measurable functions () such that
- (i)
for all ,
- (ii)
,
- (iii)
implies \sum\limits_{j\in\mathbb{N}}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}\varrho_{j,t}(x)-\varrho_{j,t}(y)\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}}<t,
- (iv)
for all and the set \mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}j\in\mathbb{N}:\operatorname{spt}\varrho_{j,t}\cap B(x,s)\neq\varnothing\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} is finite.
Here, denotes the support of a function or a measure, where the support for a measure is defined by duality with .
Theorem 2.4**.**
Let be an unbounded proper metric space of bounded geometry. Then property A is equivalent to property A′.
Proof.
Assume that has property A and fix . Let be as in Definition 2.2 with -variation. Without loss of generality we may assume that for all (otherwise consider {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}\mu_{x}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}}).
Choose and choose a sequence according to Lemma 2.1 (as is unbounded, we must have ). For let Q_{j,\delta}:=\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}x\in X:\mathrm{dist}(x,Q_{j})<\delta\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}. Then \mathcal{U}:=\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}Q_{j,\delta}:j\in\mathbb{N}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} is a countable open covering of . Let be a partition of unity subordinated to . In particular, we have for all . We define by the duality pairing . Then is continuous, hence measurable. Moreover,
[TABLE]
by monotone convergence and the partition of unity property. Therefore we have (i).
Let . By property (c) of Defintion 2.2, there exists such that is supported in for all . If with , then , so . Hence, . We get
[TABLE]
and (ii) follows.
Let with . Then, by monotone convergence and since is a partition of unity,
[TABLE]
which yields (iii).
Let and . As \mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}j\in\mathbb{N}:Q_{j}\cap B(x,s+\delta+S)\neq\varnothing\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} is finite by Lemma 2.1(e) and , (iv) follows as well.
Finally, as is assumed to be unbounded, it is clear that we need infinitely many non-zero for a partition of unity and therefore we may just delete all that are zero.
Now assume has property A*′. Let as in the definition of bounded geometry of . Let be a maximal -separated subset of and . Then is countable (and infinite since is unbounded) and discrete. Moreover, which means that is coarsely dense in . By the bounded geometry of , for there exists such that for all the ball can be covered by at most open balls (in ) of radius . Since is -separated, each of these -balls in can contain at most one point of . Thus, has at most elements. Put differently, has bounded geometry ‘with ’. Let . Let 0<t<\min\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}\tfrac{1}{R},\varepsilon\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} and as in property A′*, . Set (). Then is a cover of , and is a partition of unity of subordinated to . Let with . Then and therefore \sum_{j\in\mathbb{N}}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}\varrho_{j,t}(x)-\varrho_{j,t}(y)\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}}<t\leqslant\varepsilon. Moreover, for all . Now, [39, Theorem 1.2.4] yields that has property A. Since is coarsely dense in , [39, Lemma 5.2.4] yields that has property A. ∎
Another related notion is the asymptotic dimension. A metric space has finite asymptotic dimension if there exists such that for all there is a uniformly bounded open cover with -multiplicity less than . Here, -multiplicity means that every open ball of radius intersects with at most sets from the cover (cf. [4, Section 3]).
Proposition 2.5**.**
Let be a proper metric space of bounded geometry. If has finite asymptotic dimension, then also has property A′ (and therefore also property A).
Proof.
Assuming bounded geometry, the construction on page 6 of [18] does the job. ∎
For completeness we mention that a similar construction as in [18] was already used in [39, Corollary 2.2.11] to show that finite asymptotic dimension implies property A in the discrete setting. Note that there are proper metric spaces of bounded geometry with property A (and hence property A*′*) but infinite asymptotic dimension, see e.g. [4, Section 23].
3 Band-dominated operators on metric spaces
Let be a proper metric space of bounded geometry that satisfies property A*′*, a Borel measure on , . For every we fix a family of functions that satisfies the axioms in Definition 2.3. We will use these functions for the rest of the paper.
Lemma 3.1**.**
Let be a proper metric space of bounded geometry that satisfies property A′ and let for , and . Then
- (i)
* for all ,*
- (ii)
,
- (iii)
* implies \sum\limits_{j\in\mathbb{N}}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}\varphi_{j,t}(x)-\varphi_{j,t}(y)\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}}^{p}<t,*
- (iv)
for all and the set \mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}j\in\mathbb{N}:\operatorname{spt}\varphi_{j,t}\cap B(x,s)\neq\varnothing\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} is finite.
Proof.
(i), (ii) and (iv) are clear. For (iii) use {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}a^{1/p}-b^{1/p}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}}\leqslant{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}a-b\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}}^{1/p} for . ∎
Definition 3.2**.**
Let A\in\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)}. We call
[TABLE]
the propagation or band width of .
Let \mathrm{BO}:=\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}A\in\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)}:\;\operatorname{prop}(A)<\infty\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} denote the set of operators of finite propagation or band operators. Its norm closure \mathrm{BDO}^{p}:=\overline{\mathrm{BO}}\subseteq\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)} is called the set of band-dominated operators.
In this section we aim for various properties and a crucial characterization of . We start with a technical lemma about band operators.
Lemma 3.3**.**
Let be a proper metric space of bounded geometry (with some ) that satisfies property A′ and let . Then there is a constant such that for every , every and all band operators with the estimate
[TABLE]
holds.
Here denotes the commutator of two operators, i.e. .
Proof.
Let be a band operator with and . Let be the sequence of Borel sets coming from Lemma 2.1 with . For every choose a point . Now, define
[TABLE]
As , Lemma 3.1 implies
[TABLE]
for every and every , hence every . This implies
[TABLE]
by the monotone convergence theorem. Using the triangular inequality and Minkowski’s inequality for sums, we get
[TABLE]
It thus remains to estimate \sum\limits_{j=1}^{\infty}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}[A,M_{\tilde{\varphi}_{j}}]f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}^{p}. For this we define the relation
[TABLE]
Note that by Lemma 2.1, |\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}l\in I:k\sim l\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}|\leqslant N and that only depends on (not on or ).
As , we have
[TABLE]
Using the usual dual pairing () and Hölder’s inequality twice, we get
[TABLE]
since the are pairwise distinct and |\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}l\in I:k\sim l\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}|\leqslant N, i.e. every is counted at most times. For we have and hence
[TABLE]
by the same arguments as above. ∎
For the characterization of band-dominated operators we will need another auxiliary lemma.
Lemma 3.4**.**
Let , in , , and . Then
[TABLE]
Proof.
Let . Using the usual dual pairing and Hölder’s inequality twice, we get
[TABLE]
since . ∎
The following Proposition is an adaptation of [34, Theorem 2.1.6] to our setting. More precisely, we characterise an operator to be band-dominated by looking at (limits of) suitable commutator estimates of with multiplication operators.
Proposition 3.5**.**
Let be a proper metric space of bounded geometry that satisfies property A′ and A\in\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)}. Then the following are equivalent:
- (a)
. 2. (b)
\lim\limits_{t\to 0}\sup\limits_{{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}=1}\sum\limits_{j=1}^{\infty}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}[A,M_{\varphi_{j,t}}]f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}^{p}=0.
Proof.
“(a)(b)”: Let and choose such that {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A-A_{0}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\leqslant\varepsilon. Then, by Lemma 3.3, there is a such that \sup\limits_{{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}=1}\sum\limits_{j=1}^{\infty}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}[A_{0},M_{\varphi_{j,t}}]f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}^{p}\leqslant\varepsilon^{p} for all . It follows
[TABLE]
since . Hence \lim\limits_{t\to 0}\sup\limits_{{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}=1}\sum\limits_{j=1}^{\infty}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}[A,M_{\varphi_{j,t}}]f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}^{p}=0.
“(b)(a)”: Let , where . Then
[TABLE]
The first term vanishes since . Using Lemma 3.4, we can estimate the second term:
[TABLE]
By assumption, this tends to [math] as . Thus is a bounded net and converges to in norm. That is a band operator follows directly from the fact that . Hence . ∎
Corollary 3.6**.**
Let be a proper metric space of bounded geometry that satisfies property A′, and . Then
[TABLE]
Proof.
By Proposition 3.5 and standard commutator relations this holds for polynomials . As {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}M_{g\circ\varphi_{j,t}}-M_{p\circ\varphi_{j,t}}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\leqslant{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}g-p\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{\infty}, this generalizes to arbitrary continuous functions . ∎
Next, we show some algebraic properties of the set . These properties are well-known for band-dominated operators on -spaces over (cf. [34, Propositions 2.1.7-2.1.9]) and transfer to our situation as well.
Theorem 3.7**.**
Let be a proper metric space of bounded geometry that satisfies property A′. We have
- (a)
* for all .* 2. (b)
* is an algebra and is a closed subalgebra of \mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)}.* 3. (c)
Let be Fredholm, B\in\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)} a regulariser for . Then . In particular, is inverse closed in \mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)}. 4. (d)
* contains \mathcal{K}\bigl{(}L_{p}(X,\mu)\bigr{)} as a closed two-sided ideal.* 5. (e)
Let , . Then .
For the proof we will need the following basic fact (see e.g. [34, Theorem 1.1.3] where it is shown for sequences, but the same proof works for bounded nets).
Lemma 3.8**.**
Let be a Banach space and in be a bounded net of operators converging strongly to . Then
[TABLE]
for every . Moreover, by taking adjoints, if converges strongly to , then
[TABLE]
for every .
We say that a net converges -strongly if and both converge strongly.
Proof of Theorem 3.7.
(a) Let . We show that . Let with . Then , so . Since multiplication operators commute, we have , and therefore with .
(b) Let . It is clear that is a vector space, hence it suffices to show . Let such that . Define K_{0}:=\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}x\in X:\;\mathrm{dist}(x,K)\leqslant\operatorname{prop}(B)\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}. Then . Moreover, . Hence, . It follows that is an algebra and is a closed subalgebra of \mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)}.
(c) Assume that A\in\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)} is Fredholm, B\in\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)} and K_{1},K_{2}\in\mathcal{K}\bigl{(}L_{p}(X,\mu)\bigr{)} such that and . Fix , and for and let . Then
[TABLE]
Therefore,
[TABLE]
where we substituted g:=\frac{Bf}{{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}Bf\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}} for . By Proposition 3.5, the first term tends to [math] as .
The net converges -strongly to the identity as . Since and are compact, Lemma 3.8 implies {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}M_{\mathbbold{1}_{X\setminus B(x,R)}}K_{1}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\to 0 as well as {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}K_{2}M_{\mathbbold{1}_{X\setminus B(x,R)}}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\to 0 for . We have
[TABLE]
For the second term we observe
[TABLE]
where we used . The second term in (2) can therefore be made as small as desired by choosing large. For the first term in (2) we assume as this implies
[TABLE]
for all by Lemma 3.1. Hence
[TABLE]
We thus infer that that the second term in (3) also tends to [math] as . Similarly, we may estimate the third term in (3). Proposition 3.5 now yields the assertion.
(d) In the proof of (c) we essentially showed that
[TABLE]
for all K\in\mathcal{K}\bigl{(}L_{p}(X,\mu)\bigr{)}. Proposition 3.5 thus yields the assertion.
(e) If is a sequence in with {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A_{n}-A\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\to 0, then is a sequence in and we have {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A_{n}^{*}-A^{*}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{\mathcal{L}\bigl{(}L_{q}(X,\mu)\bigr{)}}={}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A_{n}-A\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)}}\to 0. Therefore . ∎
In what follows we will work with operators which are not necessarily defined on the whole , but only acting on closed subspaces. A particular example are Toeplitz operators.
Definition 3.9**.**
Let be a closed subspace and assume there exists a bounded linear projection P\in\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)} with . Let .
- (a)
Then is called band-dominated, provided . The set of band-dominated operators on will be denoted by . 2. (b)
Let . Then the operator is called Toeplitz-operator associated with . Let be the algebra generated by all Toeplitz operators on .
Theorem 3.10**.**
Let be a metric space that satisfies property A′and a closed subspace with projection onto . Then is a closed subalgebra of , contains , and is closed w.r.t. Fredholm inverses. In particular, is inverse closed.
Proof.
This follows directly from Theorem 3.7. ∎
4 Limit operators
In this section we turn to limit operators. We start by making three assumptions and then deduce properties of operators by means of their limit operators.
Assumption 4.1** (Space).**
Let be a proper metric space of bounded geometry (with ) that satisfies property A′. Assume that is unbounded and let be a Radon measure on . Let be the Stone-Čech compactification of , the boundary.
Assumption 4.2** (Subspaces and Projection).**
Let and let be a closed subspace with bounded projection . Moreover, assume M_{\mathbbold{1}_{K}}P,PM_{\mathbbold{1}_{K}}\in\mathcal{K}\bigl{(}L_{p}(X,\mu)\bigr{)} for all compact subsets .
Assumption 4.3** (Shifts).**
Fix . For let be a bijective isometry with . Assume that and let be a measurable function such that {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}h_{x}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}}^{p}=\frac{\mathrm{d}(\mu\circ\phi_{x})}{\mathrm{d}\mu} -almost everywhere. Assume that the maps and are continuous for -almost every . For and let be defined by and assume that extends continuously to for all compact.
These assumptions are used throughout the rest of this paper. Here are some additional remarks concerning these assumptions:
Remark 4.4**.**
- (a)
As we assume that is unbounded, cannot be compact. In fact, as is proper, assuming that is not compact is equivalent to assuming that is unbounded.
- (b)
The Radon property of guarantees that the continuous functions with compact support are dense in .
- (c)
The inverses are also isometries.
- (d)
If is proper and satisfies the symmetry assumptions in Assumption 4.3, then automatically has bounded geometry with arbitrary . Indeed, for every the closed ball is compact and hence can be convered with finitely many balls of radius . As isometries map balls to balls, the bounded geometry follows.
We proceed with a few simple lemmas which follow directly from the assumptions. For the reader’s convenience we also provide short proofs. In the following, we use the notation for the essentially bounded functions with compact support.
Lemma 4.5**.**
Let and . Then
[TABLE]
Proof.
We have
[TABLE]
Lemma 4.6**.**
Let , . Then is a surjective isometry.
Proof.
Let . Then
[TABLE]
Hence, is an isometry.
Let . Define f:=\Bigl{(}\frac{g}{h_{x}}\Bigr{)}\circ\phi_{x}^{-1}. Then and , so is surjective. ∎
Note that (U_{x}^{p})^{-1}f=\Bigl{(}\frac{f}{h_{x}}\Bigr{)}\circ\phi_{x}^{-1} for all , , and is again a surjective isometry for all .
Lemma 4.7**.**
Let , . Then .
Proof.
Let . Then for -almost every we have
[TABLE]
Proposition 4.8**.**
Let . Then the map has a weakly continuous extension to .
Proof.
We first show that the map is strongly continuous for any . Let be continuous. As and are continuous for -almost every by Assumption 4.3, is also continuous for -almost every . Moreover, {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}U_{x}^{p}f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}={}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p} for all , hence is continuous by Scheffé’s Lemma. As this is true for every continuous function and the continuous -functions are dense in , we obtain that is strongly continuous. This also implies that is strongly continuous. Indeed, if is a net in with and , then
[TABLE]
as . As a consequence, is also strongly continuous. As bounded sets are relatively compact in the weak operator topology, the map has a weakly continuous extension to . ∎
Proposition 4.8 shows that for , and every net in with the weak limit
[TABLE]
exists and is independent of the net . In particular, for .
Corollary 4.9**.**
Let . Then {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}B_{x}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\leqslant{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}B\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}} for all .
Proof.
This follows directly from the uniform boundedness principle and the fact that and are isometries. ∎
Definition 4.10** (limit operators).**
Let , and a net in that converges to . Then the operator
[TABLE]
is well-defined and called a limit operator of .
Note that is by definition an operator on the closed subspace , which may depend on and may differ from . For band-dominated operators the convergence is much stronger:
Theorem 4.11**.**
Let , compact and . For every net in converging to we have
- (i)
,
- (ii)
,
- (iii)
,
- (iv)
\operatorname*{s-lim\,}\limits_{x_{\iota}\to x}\bigl{(}(AP)_{x_{\iota}}\bigr{)}^{*}=P_{x}^{*}A_{x}^{*}.
For better readability we will divide the proof into several propositions and lemmas, which are useful and interesting in their own right.
Proposition 4.12**.**
If , then and for all .
Proof.
Let be a net in that converges to and let be compact with . Then Lemma 4.5 yields
[TABLE]
and by Lemma 4.7 we obtain
[TABLE]
for all . Taking the weak limit , we obtain and . ∎
Corollary 4.13**.**
If , then for all .
Proof.
For and there is a such that . Corollary 4.9 yields
[TABLE]
which implies by Theorem 3.7 and Proposition 4.12. ∎
Lemma 4.14**.**
Let . For all there is a such that for all and all we have
[TABLE]
Proof.
For band operators this is Lemma 3.3 combined with Proposition 4.12. For band-dominated operators we can use approximation as in Proposition 3.5 and Corollary 4.13. ∎
Lemma 4.15**.**
Let and let be a net in that converges to . Assume
[TABLE]
for all compact. Then also
[TABLE]
for all compact.
Proof.
(i) Let and compact. By Lemma 4.14 we can choose a such that for all we have
[TABLE]
Choose sufficiently large such that for . Then, using Lemma 3.4 and Corollary 4.9, we get
[TABLE]
Let be a compact set such that for all . It follows
[TABLE]
which tends to [math] by assumption. As was arbitrary, we conclude
[TABLE]
(ii) Let and compact. Again, we choose such that (4) holds and sufficiently large such that for . We have
[TABLE]
The first term tends to [math] as has compact support. The second term can be estimated using Lemma 3.4:
[TABLE]
As was arbitrary, we again conclude
[TABLE]
Corollary 4.16**.**
Let and let be a net in that converges to . Assume
[TABLE]
for all compact. Then in the strong operator topology.
Proof.
Let and . Choose compact such that {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}M_{\mathbbold{1}_{K^{c}}}f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}<\varepsilon. Then
[TABLE]
by Corollary 4.9. As was arbitrary, Lemma 4.15 yields the assertion. ∎
Lemma 4.17**.**
- (i)
* is a projection for every .*
- (ii)
* and are compact for all compact sets and .*
- (iii)
Let for . Then for every .
Proof.
First assume that . Then is a projection onto with . Also,
[TABLE]
(see Lemma 4.7) and the latter is compact by Assumption 4.2. Similarly, is compact. Moreover,
[TABLE]
and similarly .
Now let . Then there is a net in with . By Assumption 4.3 and Corollary 4.16, we have strongly. This implies that is again a projection. Also, Lemma 4.15 implies and so that and are again compact. Moreover, by Lemma 4.15, we have
[TABLE]
as . Let , , and choose compact such that {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}M_{\mathbbold{1}_{K^{c}}}g\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{q}<\varepsilon. With the usual dual pairing of and we get
[TABLE]
The first and the third term tend to [math] as by (6) and Proposition 4.8, whereas the second term is bounded by 2{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}P\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}B\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}\varepsilon. As was arbitrary, we get weakly, which implies . Similarly, we obtain . ∎
Proof of Theorem 4.11.
Let , compact and let be a net in converging to . Define . By Lemma 4.15, Corollary 4.16 and Lemma 4.17(iii), it suffices to show
[TABLE]
By Proposition 4.8, the weak limits
[TABLE]
exist and do not depend on the net converging to . By Corollary 4.9 and Lemma 4.17,
[TABLE]
The first and the second term tend to [math] by Lemma 4.15. and are compact by Lemma 4.17. Therefore and since weakly, the third term also tends to [math] as . ∎
The above may of course also be applied to . The next corollary is therefore immediate from Assumption 4.3, Theorem 4.11, Lemma 4.15 and Corollary 4.16.
Corollary 4.18**.**
Let , compact and a net in converging to some . Define . Then for all and
- (i)
,
- (ii)
,
- (iii)
,
- (iv)
.
In the next proposition we summarize a few properties of limit operators, which follow directly from properties of the strong operator convergence.
Proposition 4.19**.**
Let , a sequence in and . Then
- (a)
,
- (b)
,
- (c)
,
- (d)
{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A_{x}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\leqslant{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}},
- (e)
if in norm, then in norm.
Proof.
(a) follows from Corollary 4.13. (b), (c) and (e) follow from standard properties of strong convergence and the fact that \mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}U_{y}^{p}AP(U_{y}^{p})^{-1}:\;y\in X\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} is a bounded set. For (d) let and a net in converging to . Using that is an isometry, we get
[TABLE]
Taking the limit , we get {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A_{x}f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}\leqslant{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}. ∎
Compact Operators
In this subsection we finally show (I). One implication is straightforward to prove.
Proposition 4.20**.**
Let . Then for all . In particular, for all and .
Proof.
Let , a net in that converges to and . Then
[TABLE]
by Lemma 4.7. \bigl{(}M_{\mathbbold{1}_{B[x_{\iota},R]}}\bigr{)}_{\iota} converges -strongly to [math] as . Compactness of therefore implies
[TABLE]
as . Hence
[TABLE]
for all compact. Thus, . ∎
The reverse implication is more difficult to show and requires the following notions. For let , which is finite by Lemma 3.1 and Assumption 4.1.
Definition 4.21**.**
For A\in\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)}, a Borel set and we define
[TABLE]
These definitions are similar to [17, Definition 3.3] in case of -spaces over . They yields localized norms of the operator. The following proposition is then a version of [17, Corollary 3.5] to our case of -spaces over metric measure spaces of bounded geometry.
Proposition 4.22**.**
Let and . Then there is a such that for all , all Borel sets and all operators B\in\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}A\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}\cup\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}A_{x}:\;x\in\beta X\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} we have
[TABLE]
Proof.
Let , B\in\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}A\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}\cup\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}A_{x}:\;x\in\beta X\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} and a Borel set. Let with {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}=1 and , such that
[TABLE]
By Lemma 4.14 there is a (independent of and ) such that for all :
[TABLE]
Since for all , we have
[TABLE]
Minkowski’s inequality yields
[TABLE]
Thus, there exists such that
[TABLE]
Since for some , we get
[TABLE]
The other inequality is clear by definition. ∎
Theorem 4.23**.**
Let . Then
[TABLE]
In particular, {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A+\mathcal{K}(M^{p})\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}=\sup\limits_{x\in\Gamma X}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A_{x}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}} if {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}P\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}=1.
Proof.
Let . For we obtain by Proposition 4.20. Thus, by Proposition 4.19 we observe
[TABLE]
Hence,
[TABLE]
Concerning the first inequality, for K\in\mathcal{K}\bigl{(}L_{p}(X,\mu),M^{p}\bigr{)} we estimate
[TABLE]
Thus,
[TABLE]
It is therefore sufficient to show that
[TABLE]
Assume by contradiction that there exists such that
[TABLE]
Let be as in Assumption 4.3. Then, in particular, for all we have
[TABLE]
since is compact by Assumption 4.1 and therefore PM_{\mathbbold{1}_{B[x_{0},s]}}\in\mathcal{K}\bigl{(}L_{p}(X,\mu),M^{p}\bigr{)} by Assumption 4.3. By Proposition 4.22, there exists such that for all we have
[TABLE]
Hence, for all there exists such that
[TABLE]
Note that cannot be bounded as the second term above would be [math] for sufficiently large . Since is compact there exists a subnet of , again denoted by , and such that . By Lemma 4.6 and Lemma 4.7, we have
[TABLE]
By Theorem 4.11, \bigl{(}U_{x_{s}}^{p}AP(U_{x_{s}}^{p})^{-1}M_{\mathbbold{1}_{B[x_{0},r_{t}]}}\bigr{)} converges to . Thus,
[TABLE]
and therefore
[TABLE]
which is clearly a contradiction. ∎
By means of Theorem 4.23 we can now show the converse of Proposition 4.20.
Corollary 4.24**.**
Let . Then is compact if and only if and for all .
Proof.
If , then Theorem 3.10 implies and Proposition 4.20 (or Theorem 4.23) yields for all . On the other hand, if and for all , then Theorem 4.23 yields {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}K+\mathcal{K}(M^{p})\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}=0, i.e. . ∎
Fredholm operators
In this subsection we show (II) and (III). We start with a few preliminary results. We start with a convergence lemma (including a norm bound) which originally (i.e., for -spaces over ) goes back to Simonenko [37], see also [34, Proposition 2.2.2] and [25, Lemma 3.14].
Lemma 4.25**.**
Let and assume that the set \mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}A_{j}:j\in J\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}\subseteq\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)} is bounded. Then for every the series converges -strongly and
[TABLE]
Proof.
Let . Using Lemma 3.4, we obtain
[TABLE]
This implies the strong convergence and the norm estimate. The convergence of the adjoints is shown analogously. ∎
The next proposition is similar to [34, Proposition 2.2.3] and [25, Proposition 3.15], and relies on Lemma 4.25.
Proposition 4.26**.**
Let , , and such that for all there exists such that for all there exist B_{j,t},C_{j,t}\in\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)} with {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}B_{j,t}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}},{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}C_{j,t}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\leqslant c and
[TABLE]
Then is Fredholm and {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}(A|_{M^{p}}+\mathcal{K}(M^{p}))^{-1}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\leqslant 2{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}P\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}c.
Proof.
Let . By Lemma 4.25, the series
[TABLE]
converges strongly with {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}B_{t}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\leqslant c. Hence,
[TABLE]
For the second term we have, using Lemma 3.4,
[TABLE]
which tends to [math] as by Proposition 3.5. For the first term, we obtain
[TABLE]
Hence,
[TABLE]
as . In particular,
[TABLE]
as . As the functions have bounded support, the operators are compact for all by Assumption 4.2. Since for all , we have
[TABLE]
Therefore, by a Neumann series argument, there exists a such that and
[TABLE]
By Theorem 3.7, for . Therefore, noting that , we may apply the above to to obtain an operator
[TABLE]
with
[TABLE]
Hence
[TABLE]
using . As above, this implies that there exists a such that and
[TABLE]
Thus is Fredholm and {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}(A|_{M^{p}}+\mathcal{K}(M^{p}))^{-1}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\leqslant 2{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}P\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}c. ∎
Recall and for .
Proposition 4.27**.**
Let , a net in with , invertible and . Then there exists such that for all there exist operators B_{\iota},C_{\iota}\in\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)} with {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}B_{\iota}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}},{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}C_{\iota}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\leqslant 2({}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A_{x}^{-1}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}P\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}+{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}Q\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}) and
[TABLE]
Proof.
Let be a compact set that contains . As by Assumption 4.2, is also in . By Corollary 4.18, we have
[TABLE]
Since is invertible, is also invertible with
[TABLE]
Hence, there exists such that
[TABLE]
satisfies {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}R_{\iota}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\leqslant\frac{1}{2} for all . In particular, I+R_{\iota}\in\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)} is invertible for all . Using
[TABLE]
we get
[TABLE]
and thus
[TABLE]
Applying from the left and from the right, by Lemma 4.7, we obtain
[TABLE]
Similarly,
[TABLE]
Hence,
[TABLE]
for sufficiently large , where
[TABLE]
Theorem 4.28**.**
Let . If invertible for all and \sup\limits_{x\in\Gamma X}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A_{x}^{-1}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}<\infty, then is Fredholm.
Proof.
Assume that is not Fredholm. Note that
[TABLE]
By Proposition 4.26 there exists a and an increasing sequence in such that
[TABLE]
for all and all B\in\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)} with {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}B\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\leqslant 2\Bigl{(}\sup\limits_{x\in\Gamma X}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A_{x}^{-1}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}P\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}+{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}Q\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\Bigr{)}. W.l.o.g. (by only considering one of the two statements and choosing an approriate subsequence) we may assume that
[TABLE]
There exists such that for all by Lemma 3.1. Therefore there exist in and such that
[TABLE]
As is increasing, Lemma 3.1(iv) implies for all . Since is compact, there exists a subnet of such that for some . Let as in Assumption 4.3, i.e. for all . By Proposition 4.27 there exists such that for all there exists an operator B_{\iota}\in\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)} such that {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}B_{\iota}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\leqslant 2({}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A_{x}^{-1}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}P\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}+{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}Q\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}) and
[TABLE]
Multiplying by from the right yields
[TABLE]
for all , a contradiction. ∎
The converse of Theorem 4.28 is also true and easier to prove.
Theorem 4.29**.**
Let be Fredholm and . Then is invertible and {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A_{x}^{-1}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\leqslant{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}(A+\mathcal{K}(M^{p}))^{-1}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}.
Proof.
Let be a Fredholm regulariser of , i.e. and for two compact operators . By Theorem 3.10, . Using Proposition 4.19 and Proposition 4.20, we get
[TABLE]
and similarly, . Hence, is invertible and {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A_{x}^{-1}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}={}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}B_{x}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\leqslant{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}B\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}} by Proposition 4.19 again. As this is true for every regulariser , we get {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A_{x}^{-1}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\leqslant{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}(A+\mathcal{K}(M^{p}))^{-1}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}. ∎
Lower norm
Definition 4.30**.**
Let be complex Banach spaces and A\in\mathcal{L}\bigl{(}Y_{1},Y_{2}\bigr{)}. Then
[TABLE]
is called the lower norm of .
Lemma 4.31** (see [25, Lemma 2.35]).**
Let be complex Banach spaces and A\in\mathcal{L}\bigl{(}Y_{1},Y_{2}\bigr{)} an invertible operator. Then \nu(A)={}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A^{-1}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}^{-1}.
For our specific setup of operators on -spaces we will need two refined notions. For let as above, which is finite by assumption.
Definition 4.32**.**
For , a Borel set and A\in\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)} we define
[TABLE]
Note that .
Lemma 4.33**.**
Let and . Then .
Proof.
Let be a net in that converges to and . Moreover, choose with {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}=1 such that {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A_{x}f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}<\nu(A_{x})+\varepsilon. By Theorem 4.11 and Corollary 4.16, we can choose sufficiently large such that {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}\left(A_{x}-U_{x_{\iota}}^{p}AP(U_{x_{\iota}}^{p})^{-1}\right)f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}<\varepsilon and {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}U_{x_{\iota}}^{p}P(U_{x_{\iota}}^{p})^{-1}f-f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}<\varepsilon. Using that is an isometry, we get
[TABLE]
As was arbitrary, . ∎
Lemma 4.34**.**
Let A,B\in\mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)} and a Borel set. Then:
- (a)
{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}\nu(A|_{F})-\nu(B|_{F})\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}}\leqslant{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}(A-B)M_{\mathbbold{1}_{F}}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}. 2. (b)
For : {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}\nu_{t}(A|_{F})-\nu_{t}(B|_{F})\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}}\leqslant{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}(A-B)M_{\mathbbold{1}_{F}}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}.
Proof.
We only prove (a), the same proof also works for (b). Let . There exists with {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}=1, and {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}Bf\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}\leqslant\nu(B|_{F})+\varepsilon. Hence,
[TABLE]
Similarly,
[TABLE]
Therefore, {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}\nu(A|_{F})-\nu(B|_{F})\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}}\leqslant{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}(A-B)M_{\mathbbold{1}_{F}}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}. ∎
The next lemma is the analogue of Proposition 4.22 for the lower norm.
Lemma 4.35**.**
Let and . Then there is a such that for all , all Borel sets and all operators B\in\{A\}\cup\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}A_{x}:\;x\in\beta X\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} we have
[TABLE]
Proof.
The first inequality is clear. For the second equality let , B\in\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}A\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}\cup\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}A_{x}:\;x\in\beta X\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} and a Borel set. Let with {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}=1 and , such that
[TABLE]
By Lemma 4.14 there is a (independent of and ) such that for all :
[TABLE]
Since for all , we have
[TABLE]
Minkowski’s inequality yields
[TABLE]
Thus, there exists such that
[TABLE]
Since for some , we get
[TABLE]
As above we use the notation for .
Lemma 4.36**.**
Let , , and . Then there exists such that
[TABLE]
Proof.
We divide the proof into two steps. In step (i) we show that (7) holds up to an error of . With a compactness argument we take in step (ii).
(i) Let . Choose a net in that converges to . As is compact, we may assume that the net also converges, say to some . As is proper, we must in fact have . Indeed, otherwise for sufficiently large by
[TABLE]
This would imply , which contradicts . Choose with {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}=1, and . Now choose sufficiently large such that
[TABLE]
Note that this is possible by Corollary 4.18. Set . As and are isometries, we have {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}g\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}={}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}=1. Moreover,
[TABLE]
and
[TABLE]
i.e. .
Now let with {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}f\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}_{p}=1, and . Again, we choose , where is sufficiently large such that
[TABLE]
Since , we get
[TABLE]
Moreover, we have by the same calculation as above. Therefore we obtain .
(ii) For set in (i). This yields a sequence in with
[TABLE]
As is compact, has a subnet that converges to some . Using Lemma 4.34 and Corollary 4.18, we get
[TABLE]
Taking the limit in (8) thus yields the result. ∎
Proposition 4.37**.**
Let . Then there exists such that
[TABLE]
Proposition 4.37 has its roots in [26, Theorem 8], where the corresponding statement was shown for suitable operators on -spaces over .
Proof.
By Lemma 4.35 we obtain a sequence with and
[TABLE]
for all , all Borel sets and all B\in\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}\hat{A}_{x}:\;x\in\Gamma X\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}. Choose a sequence in with
[TABLE]
By Lemma 4.36 and (9) for , for every we can find and such that
[TABLE]
Next, using Lemma 4.36 and (9) again with , we find and such that
[TABLE]
Repeating this argument, we can find and such that
[TABLE]
for all and k\in\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}1,\ldots,n\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}.
Furthermore, by repeatedly applying the second part of Lemma 4.36, for and l\in\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}1,\ldots,n\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} we obtain
[TABLE]
where we have used that for all .
Now, let for all . As is compact, has a convergent subnet denoted by . Let us denote the limit of this subnet by . Corollary 4.18 implies {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}(\hat{A}_{y_{n_{\iota}}}-\hat{A}_{y})M_{\mathbbold{1}_{B[x_{0},4r_{t_{l}}]}}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\to 0 as . By Lemma 4.34, this also implies . Thus,
[TABLE]
Taking , we obtain \nu(\hat{A}_{y})=\inf\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}\nu(\hat{A}_{y}):\;y\in\Gamma X\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}. ∎
Theorem 4.38**.**
Let . The following are equivalent:
- (a)
* is Fredholm.* 2. (b)
For all : is invertible, and \sup_{x\in\Gamma X}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A_{x}^{-1}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}<\infty. 3. (c)
For all : is invertible. 4. (d)
For all : is invertible.
Moreover, if any of the above is true, then {}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A_{x}^{-1}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}\leqslant{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}(A+\mathcal{K}(M^{p}))^{-1}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}.
Proof.
We will prove the theorem in the following order: (a) (b) (c) (d) (b) (a). As everything is already covered above, only a few notes are necessary.
“(a) (b)”: This is Theorem 4.29.
“(b) (c)”: This is clear.
“(c) (d)”: If and is invertible, then and thus is invertible.
“(d) (b)”: Proposition 4.37 implies . As , we get \sup\limits_{x\in\Gamma X}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lVert\hbox{}\right.}}{\hbox{\textstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptstyle\left\lVert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lVert\hbox{}\right.}}}A_{x}^{-1}\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rVert}}{\hbox{\textstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptstyle\left.\hbox{}\right\rVert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rVert}}}<\infty.
“(b) (a)”: This is Theorem 4.28.
The last statement is again Theorem 4.29. ∎
By exploiting the equivalence of (a) and (c) in terms of spectral theory, we obtain the following corollary.
Corollary 4.39**.**
Let . Then
[TABLE]
5 Extension to different compactifications
In this short section we extend Theorem 4.38 and Corollary 4.39 to other compactifications of . This can be achieved rather easily due to the fact that the Stone-C̆ech-compactification has a universal property.
Proposition 5.1**.**
Let be a compactification of and the boundary. Assume and that for all and all nets in with the following limits exist
[TABLE]
Then is Fredholm if and only if is invertible for all . Moreover,
[TABLE]
Proof.
Let and choose a net in with . By compactness, there is a subnet of that converges in to some . As weak limits are unique, we have and . Hence every limit operator with respect to is also a limit operator with respect to the compactification . Conversely, every limit operator with respect to is also a limit operator with respect to . Hence, since the sets of limit operators are the same, the proposition follows from Theorem 4.38. ∎
Remark 5.2**.**
In fact, it is even possible to replace the nets by sequences. Let . Our assumptions on the measure space imply that is separable and hence the strong operator topology on \mathcal{L}\bigl{(}L_{p}(X,\mu)\bigr{)} is metrizable on bounded sets. Therefore the sequential closure of the bounded set \mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}U_{y}^{p}AP(U_{y}^{p})^{-1}:\;y\in X\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} coincides with the topological closure. In particular, every limit operator of is the strong limit of a sequence \bigl{(}U_{y_{n}}^{p}AP(U_{y_{n}}^{p})^{-1}\bigr{)}_{n\in\mathbb{N}} with . However, not every sequence tending to infinity (i.e. ) yields a limit operator as the strong limit may not exist. As the closure of \mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}U_{y}^{p}AP(U_{y}^{p})^{-1}:\;y\in X\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} is strongly compact by Theorem 4.11, every such sequence has a strongly convergent subsequence, though.
6 Applications and special cases
In this section we showcase the strength of Theorem 4.38 by providing some examples which satify our Assumptions 4.1-4.3. Some of these examples are new and others are well-known. In the latter case we therefore rediscover known results.
The trivial case
Let be a metric measure space that satisfies Assumptions 4.1-4.3 and assume that is compact. Proposition 4.20 then implies that for all . Hence every possible limit operator acts only on the null space and is therefore invertible. This implies that every operator is Fredholm. No surprise there, obviously.
Operators on
Here, we recover the classical setup for limit operators. We have with the usual metric , and the counting measure. Clearly, is proper, has bounded geometry and finite asymptotic dimension (hence property A*′* by Proposition 2.5, for an explicit construction see [25, p. 83]). Also, is a Radon measure, hence Assumption 4.1 is satisfied. Let . Then and is compact for every compact subset . This implies Assumption 4.2. Let and for all . Since is translation invariant, , and the operators are just the usual shift operators. Therefore Assumption 4.3 is satisfied as well. Thus, Theorem 4.38 recovers [26, Theorem 11] in the scalar valued case (cf. Remark 5.2).
In fact, any Radon measure with \mu(\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}x\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}})\neq 0 for all will work as all the other assumptions are trivial in this case.
Operators on
Choosing with the canonical projection yields another example. In this case the domain of a limit operator depends on . In the case , for example, we have if is a limit point at and if is a limit point at . For the situation is much more complicated as every direction in yields a different projection and hence a different domain for the limit operators.
Discrete groups
Let be an unbounded proper discrete group that satisfies property A*′* (e.g. by having finite asymptotic dimension or satisfying property A, cf. Theorem 2.4 and Proposition 2.5). Let be the counting measure and the group action with , the identity element. Choose any subset (not necessarily a subgroup) and set . is a closed subspace of and the canonical projection satisfies all assumptions. To see that extends continuously to for all compact subsets just observe that this is equivalent to extending with respect to the weak operator topology, which is always possible by Proposition 4.8.
The Fock space ()
Let with the Euclidean metric and . is proper, has bounded geometry and finite asymptotic dimension. Let be a Gaussian measure on , i.e. for some , and the closed subspace of entire functions in . By [20, Theorem 7.1] the orthogonal projection extends to a bounded projection . For we choose the usual translations given by for all . Then
[TABLE]
It is natural to choose so that is an entire function again. As is given explicitly as an integral operator, it is easy to check that , for all compact and (see e.g. [14, Proposition 7, Lemma 17]). Therefore the assumptions are again satisfied and we recover [14, Theorem 28].
Generalized Fock spaces
Choosing as above but changing the measure gives rise to more interesting examples. For example, let us choose , , and \mathrm{d}\mu(z)=\left(\frac{p}{2}\right)^{\frac{mp}{2}+1}\frac{1}{\pi\Gamma(\frac{mp}{2}+1)}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}z\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}}^{mp}e^{-\frac{p}{2}{}\mathopen{\mathchoice{\hbox{\displaystyle\left\lvert\hbox{}\right.}}{\hbox{\textstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptstyle\left\lvert\hbox{}\right.}}{\hbox{\scriptscriptstyle\left\lvert\hbox{}\right.}}}z\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right\rvert}}{\hbox{\textstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptstyle\left.\hbox{}\right\rvert}}{\hbox{\scriptscriptstyle\left.\hbox{}\right\rvert}}}^{2}}\,\mathrm{d}z, where the constant is chosen such that . Let be the closed subspace of entire functions in (the so-called Fock-Sobolev spaces [9]) and the orthogonal projection. is given as the integral operator
[TABLE]
where K(z,w)=\frac{m!}{(z\bar{w})^{m}}\Bigl{(}e^{z\bar{w}}-\sum\limits_{k=0}^{m-1}\frac{(z\bar{w})^{k}}{k!}\Bigr{)} is the reproducing kernel. As for the standard Fock spaces, extends to a bounded projection . With the usual dual pairing, we have
[TABLE]
for , and . The estimate
[TABLE]
for constants now immediately implies because it is approximated by the integral operators with kernel as . Similarly, using the Hille-Tamarkin theorem (a variant of Hilbert-Schmidt for , see e.g. [41, Theorem 41.6]) one can easily show , . Finally, we show that has a continuous extension to for all compact subsets . We have
[TABLE]
therefore we choose . It follows
[TABLE]
Define
[TABLE]
Let . Then, using Hölder’s inequality, we get
[TABLE]
as , where and are some irrelevant constants coming from the measure. Therefore we can extend continuously to by for all .
Operators on Bergman spaces
Let be a bounded symmetric domain in its standard realization as a convex circular subset of containing the origin and let be the corresponding Bergman metric. It is known that is proper, unbounded (as a metric space), has bounded geometry and finite asymptotic dimension.
Let denote the Jordan triple determinant222We refer to [11] and [12] for definitions and properties of . corresponding to and let be the standard weighted Lebesgue measure given by
[TABLE]
where and is chosen such that . It is clear that is a Radon measure. Let denote the subspace of holomorphic functions in . It is shown in [18] that for suitable values of and there is a (bounded) band-dominated projection . Moreover, it is shown that in these cases and are compact for compact subsets .
For every we can choose biholomorphic maps with and continuous for all ([19, Lemma 5]). Moreover, the standard transformation properties of show that and we can choose
[TABLE]
where denotes the genus of (see again [11, 19]). As is actually a polynomial in and , and has no zeroes in , is continuous for all as well. It remains to show that extends continuously to the Stone-Čech compactification . In [19] a differently weighted projection was used for to achieve , which trivializes the condition. Here we stick to the standard projection to show what happens. Using the explicit formula for , i.e.
[TABLE]
and the standard transformation properties of and (see again [11, 19]), we arrive at
[TABLE]
Choosing , we therefore get
[TABLE]
As is a polynomial, it extends continuously to , where is compact and is the Euclidean closure of . Moreover, as the extension has no zeroes in , it is bounded from below on . This implies that and extend continuously to the Euclidean closure of and hence to . This implies that extends continuously as well. Hence all assumptions are satisfied and we recover [19, Theorem B].
The above obviously also works for reducible domains like the polydisk. More generally, if two spaces and both satisfy the Assumptions 4.1 - 4.3, then the product space also satisfies the assumptions with the obvious choices for the measure etc.
Pluriharmonic Bergman spaces
Consider the Bergman spaces from above and replace the space of holomorphic functions by the space of pluriharmonic functions, i.e.
[TABLE]
The corresponding projection is , where
[TABLE]
and is the rank- projection onto the constant functions. As is very similar to , and the compactness of and are clear (just repeat the proofs for in [14]). Moreover, we have
[TABLE]
As above this implies that extends continuously to . As is compact, the corresponding assumptions are trivially satisfied. It follows that also satisfies the assumptions. Theorem 4.38 therefore also applies to pluriharmonic Bergman spaces.
More generally, if we have two projections that satisfy the assumptions, then also satisfies the assumptions provided that is again a projection, i.e. if . Similarly, using Lemma 4.15, satisfies the assumptions if and commute. Hence Theorem 4.38 also holds with and , respectively.
Vector valued function spaces
Assume that satisfies Assumptions 4.1 - 4.3. Since the vector valued -space
[TABLE]
is isometrically isomorphic to L_{p}(X\times\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}0,\ldots,n-1\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}},\mu\times\nu), where is the counting measure on \mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}0,\ldots,n-1\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}, we can also cover these cases here. For the metric we choose for and in X^{\prime}:=X\times\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}0,\ldots,n-1\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}. Clearly, is again a proper metric space of bounded geometry that satisfies property A*′*. Moreover, is obviously Radon. As our closed subspace we choose M^{p}\times\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}0,\ldots,n-1\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}} and our projection is P\otimes I_{\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}0,\ldots,n-1\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}}. Moreover, for and we set . Obviously, the neutral element is here. All assumptions now follow directly from the corresponding assumptions for .
Remark 6.1**.**
In order to cover also Banach space-valued -spaces with an infinite-dimensional Banach space (as well as the cases p\in\mathopen{\mathchoice{\hbox{\displaystyle\left{\hbox{}\right.}}{\hbox{\textstyle\left{\hbox{}\right.}}{\hbox{\scriptstyle\left{\hbox{}\right.}}{\hbox{\scriptscriptstyle\left{\hbox{}\right.}}}1,\infty\mathclose{\mathchoice{\hbox{\displaystyle\left.\hbox{}\right}}}{\hbox{\textstyle\left.\hbox{}\right}}}{\hbox{\scriptstyle\left.\hbox{}\right}}}{\hbox{\scriptscriptstyle\left.\hbox{}\right}}}}) we can employ the so-called -theory; cf. [34, 36]. This extension may appear elsewhere.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] N. Arcozzi, R. Rochberg, E. Sawyer, Carleson measures and interpolating sequences for Besov spaces on complex balls . Mem. Amer. Math. Soc. 182 (859), vi + 163 pp. (2006).
- 2[2] S. Beckus, D. Lenz, M. Lindner, C. Seifert, On the spectrum of operator families on discrete groups over minimal dynamical systems . Math. Z. 287 , 993–1007 (2017).
- 3[3] S. Beckus, D. Lenz, M. Lindner, C. Seifert, Note on spectra of non-selfadjoint operators over dynamical systems . Proc. Edinb. Math. Soc. 61 (2), 371–386 (2018).
- 4[4] G. Bell, A. Dranishnikov, Asymptotic Dimension . Topology and its Applications 155 (12), 1265–1296 (2008).
- 5[5] R.C. Buck, Bounded Continuous Functions on a Locally Compact Space . Michigan Math. J. 5 , 95–104 (1958).
- 6[6] C. Carvalho, V. Nistor, Y. Qiao, Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras . Electronic Research Announcements 24 , 68–77 (2017).
- 7[7] C. Carvalho, V. Nistor, Y. Qiao, Fredholm Conditions on Non-Compact Manifolds: Theory and Examples . In: C. André, M. Bastos, A. Karlovich, B. Silbermann, I. Zaballa (eds.), Operator Theory, Operator Algebras, and Matrix Theory. Operator Theory: Advances and Applications 267 , Birkhäuser, Cham, 2018.
- 8[8] S.N. Chandler-Wilde, M. Lindner, Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices . Mem. Amer. Math. Soc 210 (989), 2011.