Toeplitz Quantization on Fock Space
Wolfram Bauer, Lewis Coburn, Raffael Hagger

TL;DR
This paper extends the understanding of semi-classical limits of Toeplitz operators on Fock space, showing convergence for broader classes of symbols and providing new examples where convergence fails.
Contribution
It generalizes semi-commutator vanishing results to symbols in VMO and unbounded functions, using algebraic identities and heat transform estimates.
Findings
Semi-commutator tends to zero for symbols in VMOβ©Lβ
Characterization of symbols for which semi-commutator vanishes for all bounded functions
Examples of bounded smooth functions where semi-commutator does not vanish
Abstract
For Toeplitz operators acting on the weighted Fock space , we consider the semi-commutator , where is a certain weight parameter that may be interpreted as Planck's constant in Rieffel's deformation quantization. In particular, we are interested in the semi-classical limit \tag{}\lim\limits_{t\to 0}\|T_f^{(t)}T_g^{(t)}-T_{fg}^{(t)}\|_t. It is well-known that tends to under certain smoothness assumptions imposed on and . This result was extended to in a recent paper by Bauer and Coburn. We now further generalize this result to (not necessarily bounded) uniformly continuous functions and symbols in the algebra of bounded functions having vanishing mean oscillation on . Our approach is basedβ¦
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Toeplitz quantization on Fock space
W. Bauer
Wolfram Bauer Institut fΓΌr Analysis Welfengarten 1, 30167 Hannover, Germany
,Β
L.A. Coburn
Lewis A. Coburn Department of Mathematics SUNY at Buffalo, New York 14260, USA
Β andΒ
R. Hagger
Raffael Hagger Institut fΓΌr Analysis Welfengarten 1, 30167 Hannover, Germany
Abstract.
For Toeplitz operators acting on the weighted Fock space , we consider the semi-commutator , where is a certain weight parameter that may be interpreted as Planckβs constant in Rieffelβs deformation quantization. In particular, we are interested in the semi-classical limit
[TABLE]
It is well-known that tends to [math] under certain smoothness assumptions imposed on and . This result was recently extended to by Bauer and Coburn. We now further generalize to (not necessarily bounded) uniformly continuous functions and symbols in the algebra of bounded functions having vanishing mean oscillation on . Our approach is based on the algebraic identity , where denotes the Hankel operator corresponding to the symbol , and norm estimates in terms of the (weighted) heat transform. As a consequence, only (or likewise only ) has to be contained in one of the above classes for to vanish. For we only have to impose , e.g.Β . We prove that the set of all symbols with the property that for all coincides with . Additionally, we show that holds for all . Finally, we present new examples, including bounded smooth functions, where does not vanish.
Key words and phrases:
Toeplitz operators, Fock space, semi-commutator, semi-classical limit, heat transform, oscillation
2010 Mathematics Subject Classification:
Primary: 47B35; Secondary: 30H20, 81S10
The first and third author acknowledge support through DFG (Deutsche Forschungsgemeinschaft),
BA 3793/4-1.
1. Introduction
For a suitable family of functions and Poisson bracket , one can consider the deformation quantization (in the sense of Rieffel [18, 19]) , where is a weight parameter and consists of linear operators on an appropriate Hilbert space. Hereby essential are the following limit conditions:
[TABLE]
A typical approach to obtain such a quantization is to consider a family of weighted probability measures on some domain with corresponding Lebesgue space and then associate a Toeplitz operator to every . This construction was considered for functions on several different domains and results like (1) have been obtained under certain smoothness assumptions on the functions and (see [9, 10, 11, 12, 13, 14, 15, 16, 17]).
In this paper we are interested in the case where and is a family of Gaussian measures defined below. It has been shown in [15] that (1) holds in case and are the sum of trigonometric polynomials and -times continuously differentiable functions with compact support. In [13] this result was extended to symbols and whose derivatives up to order are continuous and bounded. It was shown recently in [5] that the second equation of (1) even holds for bounded uniformly continuous symbols (BUC).
The goal of the present paper is to push this result even further by considering (no longer bounded) uniformly continuous functions ()) as well the classical space [20] of bounded functions of vanishing mean oscillation (). We note that the algebra is contained in but is a well-studied algebra which also contains discontinuous functions.
The corresponding results for bounded symmetric domains have been obtained recently in [7] and the methods of proof are rather similar. Consider the standard identity
[TABLE]
where denotes the corresponding Hankel operator. The idea now is to show that tends to [math] if is reasonably chosen. As a benefit of this approach, we only have to assume that is uniformly bounded near [math] in order to obtain
[TABLE]
In particular, can be chosen to be an arbitrary -function.
Here is a short outline of our approach: In Section 2 we fix the notation. Let be a function of bounded mean oscillation. We generalize a norm estimate on Hankel operators in [2] to the family of Hankel operators , acting on differently weighted Fock spaces. The main issue here is to choose the constant which appears in the norm estimate independently of the weight parameter . Sections 3 and 4 contain the proof of the semi-classial limit for uniformly continuous operator symbols and bounded symbols in , respectively. In Section 5 of the paper we consider the algebra of all decomposable bounded operators acting on the direct integral of standard Gaussian weighted -spaces. An ideal is defined by
[TABLE]
Theorem 5.7 shows that the set of all symbols such that the semi-commutators
[TABLE]
for all belong to the ideal is a closed and conjugate-closed subalgebra of and it precisely coincides with .
In Section 6 we show that the first equation of (1) holds for all . In Section 7 we provide some more examples of functions that do not satisfy the second equation of (1) (a first example was already given in [5]) as well as some further comments.
2. Notation and time dependent norm estimates
Let and consider the Euclidean -space . With we denote by and the Euclidean inner product and norm, respectively. For we consider the following Gaussian probability measures
[TABLE]
and the corresponding function space on . The closed subspace of analytic functions in is denoted by and it forms a Hilbert space with reproducing and normalized reproducing kernel
[TABLE]
For a measurable function and an analytic function with , we write
[TABLE]
for the corresponding Toeplitz and Hankel operators, respectively, where denotes the orthogonal projection from onto . Note that can be expressed as the integral operator:
[TABLE]
On we have the usual inner product
[TABLE]
For any linear operator with domain and range in we have the usual operator norm . For bounded (or essentially bounded) functions we write for the supremum (or essential supremum) of .
In some of our results we allow operator symbols in the space which contains unbounded functions. Then the Toeplitz operator may be unbounded as well. Hence we have to specify the domain and carefully define operator products. This issue is discussed in [1] where a function space and an increasing scale of Hilbert spaces in are defined. It is shown that and the multiplication for all are operators acting on this scale. In particular, finite products of these operators are well-defined. One easily checks that contains the space and therefore we can form finite products of Toeplitz operators with uniformly continuous symbols.
We will also need the heat transform
[TABLE]
Remark 2.1**.**
Note that for so that by the Cauchy-Schwarz inequality. **
The mean oscillation of a function is given by
[TABLE]
We will say that a function has bounded mean oscillation if the semi-norm
[TABLE]
is finite. The (linear) space of all functions having finite semi-norm is denoted by
[TABLE]
Recall that the right hand side (as a vector space) does not depend on the parameter (see [6]) and therefore we do not indicate in the notation. A different, more standard version of is considered in Section 4. As is well-known contains unbounded functions. Hankel operators with symbols in are considered in [2] and it is shown that they are bounded on a dense domain. We mention that our notations of the spaces , and in Section 4 are different from the ones in [2].
In the remaining part of this section we generalize a norm estimate for Hankel operators in [2] to operators acting on the above family of Hilbert spaces. For each consider the operator
[TABLE]
A simple calculation shows that is an isometry onto with inverse . Moreover, restricts to an isometry from onto and for all one has:
[TABLE]
Clearly, the first equality remains true for an unbounded operator symbol when we restrict the product of operators to the maximal domain of . Combining these two relations gives
[TABLE]
The following result is Theorem 4.2 in [2]:
Theorem 2.2**.**
([2]) Let . Then the Hankel operator is bounded and there is a constant independent of such that
[TABLE]
We can use the family of isometries between the differently weighted -spaces to generalize the inequality (2.7) with a -independent constant . Note that for :
[TABLE]
and therefore we obtain the identity:
[TABLE]
Corollary 2.3**.**
Let . Then there is a constant independent of and such that
[TABLE]
Proof.
We combine Theorem 2.2 with (2.6) and (2.8):
[TABLE]
where is the constant in Theorem 2.2 which is independent of and . β
3. Operators with uniformly continuous symbols
In the following we denote by the space of all uniformly continuous complex valued functions on . Moreover, we write
[TABLE]
for the algebra of all bounded uniformly continuous functions. The next two propositions recall some results from [4]:
Proposition 3.1**.**
Let . Then is in , is Lipschitz continuous and is bounded. Moreover, the Lipschitz constant of is bounded by .
Proof.
See [4, Lemma 2.1] and [4, Proposition 3.1] for the first two statements. In order to estimate the Lipschitz constant of recall that in the proof of [4, Proposition 3.1] (see also [2, Corollary 2.7]) it was shown that for all :
[TABLE]
In the last equality we have use the identity (2.8) again. β
Proposition 3.2**.**
Let . Then uniformly on .
Proof.
See [4, Proposition 3.2]. β
Proposition 3.3**.**
Let . Then as .
Proof.
Let be fixed and choose such that for all with . We divide the domain of integration into two parts:
[TABLE]
By Proposition 3.2, we have for sufficiently small . Thus the integrand can be estimated as follows:
[TABLE]
In the case we get and thus
[TABLE]
for sufficiently small .
In the case we observe that is the sum of a bounded and a Lipschitz continuous function (cf.Β Proposition 3.1) and hence for a suitable constant . Now,
[TABLE]
for . As the integral on the right-hand side is bounded, the whole term converges to [math] as . We conclude uniformly in . Thus the assertion follows. β
Theorem 3.4**.**
Let . Then . In particular,
[TABLE]
for all or all .
Proof.
Because of , it is sufficient to show
[TABLE]
This follows from Corollary 2.3 in combination with Proposition 3.3. β
Another interesting question is: For which symbols and are the semi-commutators in (3.3) compact? In other words, what is the maximal algebra generated by Toeplitz operators so that the Calkin algebra is commutative? For BUC-symbols we will answer these questions in Proposition 3.6 below. Therefore we need the following notions:
For a bounded and continuous function (in short: ) define
[TABLE]
The space of functions having vanishing oscillation is
[TABLE]
Note that in fact . Indeed, for and we may choose a compact subset such that for all . As is uniformly continuous on compact sets, there is a so that for all with . Thus for all with .
Here is one possible way of constructing functions in :
Example 3.5**.**
Choose and compose it with a function that is continuous on , differentiable in a neighborhood of and its derivative tends to [math] as . To show that is in , choose and select such that for with . By the mean value theorem we may choose sufficiently large such that for all with and . This implies and hence as . We conclude .
For a concrete example consider .
With these preparations we can now prove the following proposition.
Proposition 3.6**.**
Let and . Then the following are equivalent:
- (i)
,
- (ii)
* and are compact,*
- (iii)
* and are compact for all .*
Proof.
Let . Then and are compact by a straight forward extension of [2, Theorem 5.3] from the case to general .
Now assume that and are both compact. Then by [2, Theorem 5.3] again and is compact by [2, Theorem 3.1]. Moreover, and thus by [3, Theorem 2.3]. As the sum of two functions in is obviously again in , we conclude that is in .
That (ii) and (iii) are equivalent is standard. β
4. Symbols in VMO
In the present section we consider the quantization problem for operators with symbols of vanishing mean oscillation. We start by recalling some notation. Consider locally integrable functions with average value
[TABLE]
on a bounded measurable subset with finite measure . We consider the variance of on
[TABLE]
as well as the corresponding quantity
[TABLE]
Definition 4.1**.**
(see [20, 21]) We say is in if the set
[TABLE]
is bounded. We say is in if is in and, for all -cubes ,
[TABLE]
If we replace by , we get new sets and .
Remark 4.2**.**
* and defined earlier are quite different. In particular, is not contained in .*
The Cauchy-Schwarz inequality shows that
[TABLE]
Direct calculation shows (eg. [8, p. 313])
Lemma 4.3**.**
We have, for arbitrary bounded measurable subsets ,
[TABLE]
so that, for ,
[TABLE]
Remark 4.4**.**
Because of Lemma 4.3, and have the additional property that -cubes can be replaced by -balls in their definitions. We need only consider the inscribed and circumscribed balls for a given cube. **
In the analysis that follows, we identify with in the standard way and we consider the set , the essentially bounded functions in . It is easy to see that . However, there are discontinuous functions in , [21, p.290]. We have:
Theorem 4.5**.**
* is a sup norm-closed, conjugate closed subalgebra of with*
[TABLE]
Proof.
A easy estimate using implies (4.1). Integrating the inequality
[TABLE]
over shows that for in , we also have in . The remainder of the proof is standard. β
Remark 4.6**.**
Theorem 4.5 gives, for (or ), a classical result of Sarason [20] for the circle.**
Now we consider the quantization problem for Toeplitz operators with bounded symbols in . In the definitions above and for convenience we pass from -cubes to Euclidean -balls
[TABLE]
centered at and with radius , cf. Remark 4.4.
Lemma 4.7**.**
Let and . Then, for all and all we have
[TABLE]
Proof.
Consider the function defined by
[TABLE]
Standard arguments show that attains a global minimum at . Hence the lemma follows from . β
Theorem 4.8**.**
Let , then .
Proof.
Let and fix a parameter which we will specify later on. According to Lemma 4.7 we can estimate the mean oscillation of as follows:
[TABLE]
We denote the first and the second integral by and , respectively, and estimate them separately. By the Cauchy-Schwarz inequality we have
[TABLE]
The first integral takes the form
[TABLE]
Therefore
[TABLE]
Since as and it follows that
[TABLE]
uniformly for . Next we estimate the second integral . We obtain
[TABLE]
Let and with (4.3) choose sufficiently large such that . With this fixed and (4.2) we can choose sufficiently small such that for all . Then
[TABLE]
and the assertion follows. β
Theorem 4.9**.**
Let , then and, in particular, we have
[TABLE]
for all and all .
Proof.
It is sufficient to show that . In fact, this follows from Theorem 4.8 and Corollary 2.3. β
5. Fock quantization algebras
We now consider the direct integral of -spaces. In the following we write decomposable operators on in the form
[TABLE]
The algebra of such operators is denoted by . The set
[TABLE]
is a closed two-sided ideal in . We also consider the direct integral of the Fock spaces
[TABLE]
Remark 5.1**.**
The Toeplitz and Hankel operators and are in for . The Toeplitz operators are also in .**
Noting that
[TABLE]
for all , we consider the set
[TABLE]
Proposition 5.2**.**
* is a closed, conjugate-closed subalgebra of and we have*
[TABLE]
Proof.
For , with , it is easy to check that . For and , we have
[TABLE]
Hence is an algebra, which clearly is invariant under complex conjugation of its elements. The equality of sets in (5.5) follows from (5.4). β
Note that the definition of the algebra and its characterization in (5.5) is given in terms of operator conditions. We now give an equivalent description which only involves a condition on the level of functions. Consider the set
[TABLE]
Proposition 5.3**.**
. In particular, is a closed, conjugate-closed subalgebra of .
Remark 5.4**.**
One may check directly from the definition of , that it is a a closed, conjugate-closed sub-algebra of . **
The next lemma serves as a preparation for the proof of Proposition 5.3. It generalizes (with the same proof) Lemma 2.5 in [2] to the family of weighted spaces and , .
Lemma 5.5**.**
Let and . Then the mean oscillation of can be represented in the form
[TABLE]
where for denotes the translation by . Moreover,
[TABLE]
Proof.
Let be fixed and note that . Therefore
[TABLE]
Using this relation and (2.2) we can express the mean oscillation in the form (5.7):
[TABLE]
Now we can prove the estimate (5.8). Note that (2.1) implies:
[TABLE]
It follows:
[TABLE]
Together with (5.7) one obtains:
[TABLE]
Now, (5.8) follows from the definition of the -semi-norm. β
Proof.
(Proposition 5.3). The inclusion directly follows from Corollary 2.3 whereas is a consequence of (5.5) in Proposition 5.2 and (5.8) in Lemma 5.5. β
Note that, by Theorem 4.5 the space is a closed, conjugate-closed subalgebra of . From Theorem 4.8 we know that
[TABLE]
Lemma 5.6**.**
There is only depending on the dimension such that for all :
[TABLE]
In particular, the following inclusion holds:
Proof.
It is easy to check (see [8]) that
[TABLE]
where for we define:
[TABLE]
Hence, by applying Lemma 4.3, we can estimate the mean oscillation from below as follows:
[TABLE]
Note that there is a constant only depending on the complex dimension such that
[TABLE]
From this, the statement follows. β
Finally, (5.9) together with Lemma 5.6 shows:
Theorem 5.7**.**
.
Theorem 5.7 indicates that in case of the ideal the algebra plays a similar role as in case of compact operators (cf. Proposition 3.6).
6. On the limit of the norm of Toeplitz operators
For each function in , [15, Theorem 1] shows the following identity
[TABLE]
Here, we extend this result to operator symbols by showing that converges pointwise almost everywhere to . The result will then be a consequence of Remark 2.1.
Recall that the heat transform \big{(}\tilde{f}^{(t)}\big{)}_{t>0} has the semi-group property, i.e.Β for and it holds
[TABLE]
Moreover, the assignment is a contraction on , i.e.Β for every . Letting and combining these properties shows
[TABLE]
Therefore is monotone decreasing and exists for all .
As a further preparation, we show that the heat transform is bounded by the Hardy-Littlewood maximal function for any locally integrable . The Hardy-Littlewood maximal function is defined by
[TABLE]
where denotes the (Lebesgue) volume of the ball with radius as usual.
Lemma 6.1**.**
For every , and any locally integrable function we have , where is some constant that only depends on the dimension .
Proof.
Recall that the volume of a (real) -dimensional ball of radius is given by . The result now follows by the following computation:
[TABLE]
Now we are ready to prove the theorem announced at the beginning of this section.
Theorem 6.2**.**
Let . Then for almost every . In particular,
[TABLE]
Proof.
Let , compact and . Choose sufficiently large such that and
[TABLE]
for all and . To see that this is possible observe that for sufficiently large the weight is decreasing as . We may thus choose a radius such that (6.11) holds for some . Adding the diameter of to yields a sufficiently large radius .
Cutting at that radius, i.e.Β setting , we get
[TABLE]
for all and . Now choose a continuous function of compact support such that . Then, by the inequality above and Proposition 3.2,
[TABLE]
for all and sufficiently small . To obtain the assertion, we need to show that
[TABLE]
is a null set for all .
By Lemma 6.1, we have
[TABLE]
and as is well-known, the Hardy-Littlewood maximal function satisfies the weak (1,1)-inequality, i.e.Β there exists a constant depending only on the dimension such that
[TABLE]
for all (see [22, Theorem 1]). Applying this to , we obtain
[TABLE]
Moreover,
[TABLE]
by Markovβs inequality. Thus
[TABLE]
Since was arbitary, we get
[TABLE]
for all and all compact sets . Clearly, by taking a countable covering, this remains true if we replace by in the above formula. Therefore converges to for almost every .
To prove the second assertion choose for every a bounded set with such that for all . By Egorovβs theorem, we can additionally assume uniformly for all . It follows
[TABLE]
for all . Using Remark 2.1 and the obvious inequality , we conclude
[TABLE]
7. Examples
In this section we provide three explicit examples. The first two are counterexamples to the statements of Theorem 3.4 and Theorem 4.9. Clearly, these functions cannot be uniformly continuous or have vanishing mean oscillation. The third example shows that the semi-commutator of two unbounded Toeplitz operators can be bounded and that its norm can still tend to [math] as .
(A): Direct calculation shows that a natural orthonormal basis for consists of the functions
[TABLE]
Moreover, for and , we see that are diagonal in the orthonormal basis with eigenvalues
[TABLE]
Since ,
[TABLE]
for all . We also observe that .
(B): Here is another counterexample to the quantization result in Theorem 3.4 and Theorem 4.9. Different from the example in (A) we choose a symbol which has high oscillation inside the domain (in a zero-neighbourhood). Such an effect was already observed in [5], however, the present example is even simpler. Let and consider the following symbol:
[TABLE]
We have for all . Let be the family of isometries defined in (2.4). As was observed in (2.5) the Toeplitz operator transforms under conjugation by as follows:
[TABLE]
Choose a sequence where . Then we have
[TABLE]
Because of we obtain:
[TABLE]
One can check (e.g. using the fact that is a diagonal operator) that is non-zero. Since () also does not depend on we cannot have .
(C): We consider the annihilation and creation operators on the one-particle bosonic Fock space for all . Standard calculation shows for :
[TABLE]
It follows for that
[TABLE]
Noting that yields
[TABLE]
Remark. Direct computational checks of Theorem 3.4 in the diagonal case bring us to (or even over) the edge of what is possible using Stirlingβs approximation. For an example, consider the estimation of
[TABLE]
Acknowledgement: We thank Jingbo Xia for his useful conversation and comments. In particular, he pointed out the simple proof of Corollary 2.3 which replaced our previous (and slightly weaker) estimate on the norm of the Hankel operator .
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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