Spectral dissection of finite rank perturbations of normal operators

TL;DR

This paper investigates how finite rank perturbations affect the spectral properties of normal operators using a functional model based on a perturbation matrix, revealing conditions for invariant subspaces and decomposability.

Contribution

It introduces a functional model for finite rank perturbations of normal operators that encodes spectral behavior via a perturbation matrix, providing new insights into invariant subspaces.

Findings

Operator T admits invariant subspaces under certain conditions.

Operator T can be decomposable with mild spectral measure assumptions.

Functional model links spectral behavior to a perturbation matrix.

Abstract

Finite rank perturbations $T=N+K$ of a bounded normal operator $N$ on a separable Hilbert space are studied thanks to a natural functional model of $T$; in its turn the functional model solely relies on a perturbation matrix/ characteristic function previously defined by the second author. Function theoretic features of this perturbation matrix encode in a closed-form the spectral behavior of $T$. Under mild geometric conditions on the spectral measure of $N$ and some smoothness constraints on $K$ we show that the operator $T$ admits invariant subspaces, or even it is decomposable.

Equations

$$T=N+\sum _{k=1}^m{u}_j\otimes v_j=N+\sum _{k=1}^m{u}_j⟨\cdot ,v_j⟩,$$

$$Nf(z)=zf(z)\text{on }\mathcal{H}=L^2(\mu ).$$

$$T=M_z+u(z)⟨\cdot ,v⟩:L^2(\mu )⟶L^2(\mu ),$$

$$\psi (z)=1+\int \frac{u(w)\overline{v(w)}d\mu (w)}{w-z},$$

$$\int \frac{x(w)\overline{v(w)}d\mu (w)}{w-z}+b(z),z\in \Omega ,$$

$$\Vert x{\Vert }_{2,\mu }^2+\Vert b{\Vert }_{2,\partial \Omega }^2.$$

$$x\mapsto \int \frac{x(w)\overline{v(w)}d\mu (w)}{w-z}$$

$$\mathcal{D}=\bigcup _{k\in \mathbb{Z}}{\mathcal{D}}_k.$$

$${\int }_0^1{t}^{-2}h(t)dt<\infty .$$

$$h(t)=t^a,$$

$$\nu (Q)\le h(|Q|),$$

$${\mathcal{G}}_s(\nu )=\bigcup _{Q\text{dyadic and heavy},|Q|\le s}10Q^o,\text{where }s>0.$$

$$\underset{s\to 0^{+}}{\lim }{\mathcal{H}}_{\mathrm{cont}}^1({\mathcal{G}}_s(\nu ))=0.$$

$$\sum _{Q\text{dyadic and heavy}}|Q|<\infty$$

$$\nu=\big{(}\sum_{j}|u_{j}|^{2}+|v_{j}|^{2}\big{)}\mu.$$

$$C(\tau )(z):=\int \frac{d\tau (t)}{t-z}$$

$$|C(f\nu )(z)|\le K{s}^{-1}\Vert f{\Vert }_{\infty },z\in \mathbb{C}\setminus {\mathcal{G}}_s(\nu ),$$

$$C_{\partial\Omega}(w):=C(w\cdot dz\big{|}_{\partial\Omega}),$$

$$\underset{n}{\sup }{\int }_{\partial {\Omega }_n}|f(z){|}^p|dz|<\infty .$$

$$L^2(\partial \Omega )=E^2(\Omega )\dotplus E_0^2({\Omega }^c)$$

$$\mathrm{Carleson}(E,\mu ):=\underset{z\in E,r>0}{\sup }\frac{\mu (B(z,r))}{r},$$

$$J_{\Omega }:E^2(\Omega )\to L^2(\mu ),J_{\Omega }f:=f.$$

$$\nu(Q)\leq\mu(Q)^{1/q}\bigg{(}\int_{Q}|u|^{p}\,d\mu\bigg{)}^{1/p}\leq\|u\|_{p}|Q|^{2/q},$$

$$Nf(z)=zf(z),f\in \mathcal{H}=L^2(\mu ,H):={\int }^{\oplus }H(z)d\mu (z),$$

$$T_0:=T|{\mathcal{H}}_0$$

$${\mathcal{H}}_0:=L^2(\mu ,H_0).$$

$$\big{(}x\odot y\big{)}(z):=[x(z),y(z)]_{H_{0}(z)}$$

$$[x(z),y(z){]}_{H_0(z)}=\sum _{j=1}^{n(z)}{x}_j(z)y_j(z).$$

$$⟨x(z),y(z){⟩}_{H_0(z)}=[x(z),\bar{y}(z){]}_{H_0(z)}.$$

$$\sum _{j=1}^m⟨\cdot ,v_j⟩u_j=u^t⟨\cdot ,v⟩.$$

Full text

The authors acknowledge partial support by Spanish Ministry of Science, Innovation and Universities (grant no. PGC2018-099124-B-I00), the ICMAT Severo Ochoa project SEV-2015-0554 of the Spanish Ministry of Economy and Competitiveness of Spain and the European Regional Development Fund. The first author expresses his gratitude to the Universidad Autonoma de Madrid for hospitality. We are both grateful to the Mittag-Leffler Institute for offering the opportunity to progress on this project.

1. INTRODUCTION

Finite rank perturbations of Hilbert space operators were studied for at least a century, for instance for their far reaching connections with function theory of a complex variable, boundary value problems of mathematical physics or for applications to quantum theory. It is sufficient to mention the case of dissipative operators with rank-one imaginary part or the celebrated phase-shift and related perturbation determinant, see [22] for the golden era references, or [44] for more recent developments. The booming topics of Aleksandrov-Clark measures [38] and the resurrection of Aronszjan-Donoghue theory for matrix valued measures associated to finite rank perturbations of self-adjoint operators [33, 21] are two other notable examples. To name only one less known, additional relevant ramification: an apparently non-related open problem of approximation theory, known as Sendov conjecture, can be translated into spectral estimates of rank-two perturbations of normal matrices, see [28].

Let $\mathcal{H}$ be a complex, separable Hilbert space. The class of operators under study is

[TABLE]

where $N$ is a normal operator on $\mathcal{H}$ and the finite rank summand is subject to certain “smoothness” conditions. To fix ideas we first assume that $N$ has spectral multiplicity $1$. According to the Spectral Theorem, $N$ is unitarily equivalent to the multiplication operator by the complex variable:

[TABLE]

In this functional model setting the announced “smoothness” properties of the functions $u_{j}$, $v_{j}$ will be quite natural, for instance imposing their uniform boundedness. The case of general spectral multiplicity of $N$ is covered by our main results, with notation and conventions described in Section 3.

The second author has already treated smooth trace class perturbations in the case of an absolutely continuous spectral measure (with respect to area measure) [49]. A natural quotient model of $T$, defined in terms of certain Sobolev-type spaces was introduced there. The present article builds on [49] by proposing a quotient functional model for a wide class of spectral measures $\mu$. The concrete function theoretic features of the quotient model provide the announced spectral decomposition results. In the present article we only treat finite rank perturbations due to inherent technical complications.

To give a preview of the nature of the functional model we propose in this work, we consider the simplest case

[TABLE]

with bounded measurable functions $u,v$ defined on the support $\sigma$ of the positive measure $\mu$. Let $\Omega$ be a domain with smooth boundary containing $\sigma$, so that the perturbation function

[TABLE]

does not vanish outside $\Omega$. Define the model space ${\rm Mod}(\Omega)$ of Cauchy transforms plus analytic functions

[TABLE]

where $x\in L^{2}(\mu)$ and $b(z)$ is analytic in the associated Hardy space $H^{2}(\Omega)$. This space is complete with respect to the Sobolev type norm

[TABLE]

The reader can immediately verify that the operator

[TABLE]

intertwines $T$ and multiplication by the complex variable on the quotient space ${\rm Mod}(\Omega)/\psi{\rm Mod}(\Omega).$ This precise similarity transform allows us to infer spectral decomposition properties of $T$ from function and measure theoretic results.

Our inquiry is affiliated to a series of recent developments. For instance, in a series of papers [18]–[20], Foiaş, Jung, Ko and Pearcy considered rank one perturbations of normal operators with a discrete spectral measure $\mu$. Their technique of cutting the spectrum of the perturbation relied on carefully chosen integration contours applied to localized resolvent, consequently producing appropriate Riesz projections. Later, their results were extended by Fang and Xia [16], and by Klaja [29] to finite rank and compact perturbations. Klaja also has some results [29] for a non-discrete measure $\mu$, provided there is a Jordan curve $\Gamma$ with certain continuity and growth conditions on the local resolvents $(N-\lambda)^{-1}u_{j}$ and $(N^{*}-\bar{\lambda})^{-1}v_{j}$, where $\lambda\in\Gamma$.

An earlier work on the same subject is [26]. And even before, studies of similar flavor go back to [9, 43]; see also the references in [49].

In [30], Klaja proves that if $N$ is diagonalizable and its spectrum is a perfect set, then $N$ possesses a rank one perturbation without eigenvalues. If $N$ is compact and self-adjoint, this is no longer true in general; the criterion for the existence of rank one perturbation of this type was given in [4].

On a related, but totally different line of research, weakly convergent integrals of localized resolvents of operators belonging to a $II_{1}$ factor appeared in the works of Haagerup and Schultz [24] and followers [14]. These integrals provided Riesz type projections and ultimately unveiled a rich spectral decomposition behavior.

Rank-one perturbations of normal operators were intensively studied during the last years by Baranov and his collaborators (see [1, 3]), in particular, by Baranov and the second author [4]–[6]. These works make use of a de Branges type functional model and heavily rely on the theory of entire functions.

By far from being exhaustive, more references to old and new analyses of finite rank perturbations of linear transforms are scattered through the present article.

Our aims are:

• •

To find conditions for possibly “dissecting” the spectrum of $T$ along a curve. Given a domain $\Omega$ containing $\sigma(T)$, such that ${\overline{\Omega}}={\overline{\Omega}}_{1}\cup{\overline{\Omega}}_{2}$, the domains $\Omega_{1}$ and $\Omega_{2}$ are disjoint and have piecewise smooth boundaries, our Theorem 3.9 asserts that, under additional conditions, there exists a direct sum decomposition $T=T_{1}\dotplus T_{2}$, satisfying the spectral localization conditions $\sigma(T_{j})\subset\overline{\Omega}_{j}$.

• •

To deduce from this fact sufficient conditions for the existence of invariant subspaces.

• •

To give sufficient conditions for the decomposability of $T$ in the sense of Foiaş [17].

While we do not seek in the present article the most general conditions imposed on the spectral measure $\mu$ and/or the finite rank perturbations, our approach gives rise to challenging questions in geometric function theory. Some of these problems are summarized in the last section.

The contents is the following. Section 2 contains some preliminaries of geometric measure theory and function theory of a complex variable. There we introduce the concept of dissectible Borel measure to become the key technical tool for the rest of the article. Section 3 is devoted to the construction of the Sobolev space type functional model for the perturbed operator. Our main spectral dissection theorem is stated in Section 3. In Section 4 we provide proofs for the geometric measure theory lemmas and in Section 5 we turn to the proofs involving the functional model. Section 6 is focused on the existence criterion of a non-trivial invariant subspace for the perturbed operator. In Section 7 we recall some basic terminology and facts from local spectral theory; there we propose the notion of dissectible operator, slightly stronger than decomposable operator. In the same section we provide sufficient criteria for the perturbed operator to be either dissectible, or only to possess Bishop’s property $(\beta)$. In Section 8 we formulate a few open questions.

We dedicate this article to Dan Virgil Voiculescu whose exceptional insight into perturbation theory of linear operators has reformed modern mathematical analysis. His contributions to the subject span more than four decades, to cite only [47, 48].

2. PRELIMINARIES

This section collects a few function and measure theoretic results referred to in the sequel. We also introduce some necessary notation and terminology.

Henceforth we rely on the standard dyadic system $\mathcal{D}$ of squares in $\mathbb{C}$ with sides parallel to coordinate axes. The $k$th generation $\mathcal{D}_{k}$ consists of squares of the form $[m2^{-k},(m+1)2^{-k})\times[n2^{-k},(n+1)2^{-k})$; here $k,m,n\in\mathbb{Z}$. Then

[TABLE]

A square $Q$ is dyadic if $Q\in\mathcal{D}_{k}$ for some $k$; in this case we set $|Q|=2^{-k}$ to be its side length.

For any dyadic $Q\in\mathcal{D}$ and any $c>0$, we denote by $\overline{Q}$ the closure of $Q$, by $Q^{o}$ its interior and by $cQ$ the square homothetic to $Q$ with the same center, such that $|cQ|=c|Q|$.

We denote by $\mathcal{H}^{s}$ the $s$-dimensional Hausdorff measure and by ${\mathcal{H}}^{s}_{\rm cont}$ the $s$-dimensional Hausdorff content, calculated with respect to the dyadic system $\mathcal{D}$.

Definition 2.1**.**

Throughout this article $h$ stands for a nonnegative increasing function defined on $[0,+\infty)$ satisfying $h(t)>0$ for $t>0$, $\lim_{\infty}h=+\infty,$ and

[TABLE]

Such a function $h$ is called a scale function.

The main example is

[TABLE]

where $1<a\leq 2$. To capture finer effects (related to measures “close” to the arc length measure in the sense of dimension), one can also consider functions of the form $h(t)=t\cdot\log^{-a}(2+1/t)$, $a>1$.

Definition 2.2**.**

Let $\nu$ be a finite positive measure on $\mathbb{C}$ with compact support.

1. We say that a dyadic square $Q$ is light (w.r. to $\nu$) if

[TABLE]

and heavy in the opposite case.

2. Given a measure $\nu$, consider the increasing family of open sets

[TABLE]

We say that $\nu$ is dissectible if

[TABLE]

In this case, we will also use the terms $h$-dissectible, or $a$-dissectible, for the case $h(t)=t^{a}$.

Notice that condition

[TABLE]

is sufficient for $\nu$ to be $h$-dissectible.

We remark that for any finite measure $\nu$, every sufficiently large square in $\mathcal{D}$ is light. This implies the following observation. Suppose $\nu$ is a $h$-dissectible measure as above. Choose any function $h_{1}$ meeting the same conditions as $h$ and such that $h_{1}(t)/h(t)\to\infty$ as $t\to 0^{+}$. Then for any nonnegative function $f\in L^{\infty}(\nu)$, $f\nu$ is $h_{1}$-dissectible.

In what follows, we set

[TABLE]

Most of our results require the following condition:

1. (D)

The measure $\nu$ is dissectible.

Given a complex measure $\tau$ of finite total variation, we put

[TABLE]

to be its Cauchy transform. It is known that $C(\tau)$ is defined at least as an element of $L^{1}_{\text{loc}}(\mathbb{C})$.

As shown by Verdera [46] and by Mattila and Melnikov [34], for any measure $\nu$ and any Ahlfors-David regular curve $\gamma$, the Cauchy integral $C(\nu)$ exists a.e. on $\gamma$ and satisfies a weak $L^{1}$ estimate on $\gamma$. Our constructions below require a stronger regularity of the Cauchy integrals $C(f\nu)$. In particular, we will use the following lemma, which reveals the importance of dissectible measures in our study.

Lemma 2.3**.**

For any positive measure $\nu$, any $s\in(0,1]$ and any $f\in L^{\infty}(\nu)$, one has

[TABLE]

where $K$ is a constant only depending on $\nu$. Moreover, $C(f\nu)$ is a continuous function on $\mathbb{C}\setminus\mathcal{G}_{s}(\nu)$.

In particular, if $\nu$ is a dissectible measure, then $C(f\nu)$ is continuous on sets with “small” complement.

If $\Omega$ is a Jordan domain with piecewise $C^{1}$ smooth boundary, we will use the abbreviation

[TABLE]

(here $dz$ corresponds to the positive orientation of $\partial\Omega$). We adopt the notation $L^{p}(\partial\Omega)=L^{p}(\partial\Omega,|dz|)$.

If $\Omega$ is a Jordan domain in the complex plane, by an exhaustion of $\Omega$ we mean an increasing sequence of Jordan domains $\Omega_{n}$ with rectifiable boundaries such that $\cup_{n}\Omega_{n}=\Omega$. For $1\leq p<\infty$, $E^{p}(\Omega)$ denotes the Smirnov space of analytic functions in $\Omega$. We recall that a function $f$, holomorphic in $\Omega$, belongs to $E^{p}(\Omega)$ if there is an exhaustion $\{\Omega_{n}\}$ of $\Omega$ such that

[TABLE]

These are Banach spaces for any $p$ as above; for $p=2$, they are Hilbert spaces. The space $E^{\infty}(\Omega)=H^{\infty}(\Omega)$ is just the space of bounded analytic functions on $\Omega$.

We recall some basic properties of Smirnov spaces (see [13, 52]). Denote by $K(\partial\Omega)$ the David constant of the curve $\partial\Omega$, that is, the least constant such that $\mathcal{H}^{1}(\partial\Omega\cap B(z,r))\leq Kr$ for all disks $B(z,r)$ in the plane. If $\partial\Omega$ is sufficiently good (say, locally Lipschitz), then one can use the same exhaustion $\{\Omega_{n}\}$ in (7) for all functions $f$. Namely, take any exhaustion $\{\Omega_{n}\}$ of $\Omega$ with uniformly bounded David constants $K(\partial\Omega_{n})$ (such a sequence of domains always exists). Then a function $f$, holomorphic in $\Omega$, belongs to $E^{p}(\Omega)$ if and only if (7) holds for this sequence of domains.

Let $\Omega^{c}=\widehat{\mathbb{C}}\setminus{\overline{\Omega}}$ be the complementary domain, where $\widehat{\mathbb{C}}$ is the Riemann sphere. For a function $f$ belonging to $E^{2}(\Omega)$ or to $E^{2}(\Omega^{c})$, its boundary values on $\partial\Omega$ are well-defined as an element of $L^{2}(\partial\Omega)=L^{2}(\partial\Omega,|dz|)$. In this sense, the spaces $E^{2}(\Omega)$ and $E^{2}(\Omega^{c})$ can be identified with closed subspaces of $L^{2}(\partial\Omega)$. The direct sum decomposition

[TABLE]

holds; here $E^{2}_{0}(\Omega^{c})=\{f\in E^{2}(\Omega^{c}):f(\infty)=0\}$. In general the above direct sum is not orthogonal. However, the parallel projections onto the two direct summands in (8) are given by the linear bounded transformations f\mapsto C_{\partial\Omega}f\big{|}_{\Omega} and f\mapsto-C_{\partial\Omega}f\big{|}_{\Omega^{c}}.

Definition 2.4**.**

Given a measure $\mu$ on $\mathbb{C}$ and a subset $E\subset\mathbb{C}$, define the corresponding Carleson constant of $E$ with respect to $\mu$ by

[TABLE]

where $B(z,r)$ is the open disc in $\mathbb{C}$ of radius $r$, centered at $z$.

Suppose $\Omega$ is a domain satisfying $\mu(\partial\Omega)=0$. Consider the Carleson’s embedding operator:

[TABLE]

It is known [13, 52] that both operators $J_{\Omega}$ and $J_{\Omega^{c}}$ are bounded if and only if Carleson’s constant of the boundary $\partial\Omega$ with respect to $\mu$ is bounded.

Definition 2.5**.**

A Jordan curve $\gamma$ is called admissible (with respect to $\mu$) if it satisfies the following conditions:

1. (1)

$\gamma$ is a piecewise $C^{1}$ smooth curve; 2. (2)

Let $\Omega$ and $\Omega^{c}$ be the two connected components of $\widehat{\mathbb{C}}\setminus\gamma$. There exists an exhaustion $\{\Omega_{n}\}$ of the domain $\Omega$ such that all curves $\gamma$ and $\partial\Omega_{n}$ are contained in $\mathbb{C}\setminus\mathcal{G}_{s}(\mu)$ for some (fixed) $s>0$ and all David constants $K(\partial\Omega_{n})$ as well as Carleson constants $\operatorname{Carleson}(\partial\Omega_{n},\mu)$ are uniformly bounded. The same property holds for $\Omega^{c}$.

If $\gamma$ is admissible, we simply call $\Omega$ and $\Omega^{c}$ admissible domains.

Remark 2.6**.**

It is easy to see that $\mu(\gamma)=0$ whenever the Jordan curve $\gamma$ does not intersect $\mathcal{G}_{s}(\mu)$ for some $s>0$.

Lemma 2.7**.**

Let $\mu$ be any compactly suppported finite measure on $\mathbb{C}$. Then there are subsets  $\mathbb{A}_{x},\mathbb{A}_{y}$  of the real line such that $\mathcal{H}^{1}(\mathbb{R}\setminus\mathbb{A}_{x})=\mathcal{H}^{1}(\mathbb{R}\setminus\mathbb{A}_{y})=0$ possessing the following properties:

1) Any Jordan broken line $\gamma$ is admissible whenever it consists of finitely many intervals parallel to the axes and its vertices belong to $\mathbb{A}_{x}+i\mathbb{A}_{y}$.

2) For any $x_{0}\in\mathbb{A}_{x}$, there is an increasing sequence $\{x_{n}^{-}\}$ and a decreasing sequence $\{x_{n}^{+}\}$ both tending to $x_{0}$ and such that for some $s>0$, all the points $x_{0},x_{n}^{-}$ and $x_{n}^{+}$ belong to $\mathbb{A}_{x}\setminus\operatorname{Re}\mathcal{G}_{s}(\mu)$. A similar approximation property holds for the set $\mathbb{A}_{y}$ (with $\mathbb{A}_{y}\setminus\operatorname{Im}\mathcal{G}_{s}(\mu)$ replacing $\mathbb{A}_{x}\setminus\operatorname{Re}\mathcal{G}_{s}(\mu)$).

Examples 2.8**.**

1. Let $\mu$ be the area measure restricted to some bounded Borel subset of $\mathbb{C}$. Let $d\nu=u\,d\mu$, where $u\in L^{p}(\mu)$. Let $1/p+1/q=1$. If $p>2$, then by Hölder’s inequality, any dyadic square is light:

[TABLE]

and $a=2/q>1$. So each measure $\nu$ of this form is $a$-dissectible (as it was mentioned, the constant $\|u\|_{p}$ does not matter).

Similarly, assume that $\mu$ satisfies an estimate $\mu(Q)\leq C|Q|^{\sigma}$ for any square $Q$, where $\sigma\in[1,2)$ and $C$ are constants. Choose $q\in(1,\sigma)$ and the corresponding $p\in(1,\infty)$. Then that a measure $d\nu=u\,d\mu$, where the density $u$ is in $L^{p}(\mu)$, is $a$-dissectible for $a=\sigma/q>1$. This is obtained in the same way as in the previous example.

3. On the opposite side, if the closed support $\operatorname{supp}\mu$ has Minkowski dimension less than $1$, then it is easy to see that $\mu$ is $a$-dissectible for any $a>1$ (a square can be heavy only if it touches $\operatorname{supp}\mu$).

4. Of course, the closed support is a very rough characteristic of “the dimension” of $\mu$. Suppose, however, that measures $\mu_{k}$, $k\geq 0$, are mutually singular and $h_{k}$-dissectible, for some scale functions $h_{k}$. Suppose $\sum\mu_{k}(\mathbb{C})$ is finite. Put $\mu=\sum\mu_{k}$. This measure can fail to be dissectible, but there is an equivalent measure to $\mu$ that is $h$-dissectible for some $h$. Namely, the following Proposition holds.

Proposition 2.9**.**

Suppose that measures $\mu_{k}$ are $h_{k}$-dissectible for some scale functions $h_{k}$, and suppose that the union of the closed supports of these measures is a bounded subset of $\mathbb{C}$. Then there are (small) positive constants $c_{k}$ such that the measure $\sum_{k}c_{k}\mu_{k}$ is finite and dissectible with respect to some scale function $h$.

5. In particular, for any atomic measure $\mu=\sum\delta_{t_{k}}$ there is an equivalent measure $\nu=\sum c_{k}\delta_{t_{k}}$, which is dissectible. This can be related to the techniques proposed by Foiaş and his collaborators [18]–[20].

The proofs of Lemmas 2.3,  2.7 and Proposition 2.9 will be given in Section 4.

Remark 2.10**.**

A measure $\nu$ is not $h$-dissectible (for any scale function $h$) whenever for some Ahlfors-David curve $\gamma$, $\nu|\gamma$ has a nontrivial part, absolutely continuous with respect to arc length. This observation can be deduced from Lemma 2.3.

One can infer from the above examples that measures of “dimension” $1$ are the most difficult case, at least for this approach.

3. THE FUNCTIONAL MODEL

We embark now on the higher spectral multiplicity framework. Specifically, the Spectral Theorem implies that by passing to a unitarily equivalent operator, one can assume that the normal operator $N$ is represented by a von Neumann direct integral:

[TABLE]

where $\{H(z)\}$ is a measurable family of Hilbert spaces, referred to as fibers.

If the fiber space is constant, that is, $H(z)\equiv L$ $\mu$-a.e., then $\mathcal{H}=L^{2}(\mu)\otimes L$. This is why we adopt the shorter notation $L^{2}(\mu,H)$ for the direct integral in (10).

Put $H_{0}(z)=\operatorname{span}\{u_{j}(z),v_{j}(z):1\leq j\leq m\}$; then $H_{0}(z)\subset H(z)$ and $\dim H_{0}(z)\leq 2m$ $\mu$-a.e. We make the following second standing assumption:

1. (K)

For $\mu$-almost every $z\in\mathbb{C}$, the vectors $v_{j}(z)$, $j=1,\dots,m$ generate the space $H_{0}(z)$.

Any finite rank perturbation can be rewritten in a form that satisfies (K). Indeed, it suffices to add to the expression (1) finitely many new formal terms $\langle\cdot,v_{j}\rangle\,u_{j}$ with $u_{j}=0$.

From now on, we build the quotient functional model only for the restriction

[TABLE]

of $T$ to its reducing subspace

[TABLE]

Notice that the restriction of $T$ to $\mathcal{H}\ominus\mathcal{H}_{0}$ is normal and is already represented in its von Neumann functional model.

Let $x,y\in\mathcal{H}_{0}$ be a pair of vectors (so that $x(z),y(z)$ are measurable cross-sections, $x(z),y(z)\in H_{0}(z)$, and $\|x(\cdot)\|,\|x(\cdot)\|\in L^{2}(\mu)$). Their $\odot$ product is defined by:

[TABLE]

(so that $x\odot y\in L^{1}(\mu)$). To simplify notation, we assume henceforth that the products $[x(z),y(z)]_{H_{0}(z)}$ are bilinear. To be more precise, fix a measurable family of orthonormal bases $e_{j}(z)$, $1\leq j\leq n(z)$ in spaces $H_{0}(z)$ and define these products in terms of corresponding coordinates:

[TABLE]

The usual sesquilinear product in a fiber $H_{0}(z)$ (used in (1)) is given by

[TABLE]

Here $\bar{y}(z)=\sum_{j}\bar{y}_{j}(z)e_{j}(z)$ whenever $y(z)=\sum_{j}y_{j}(z)e_{j}(z)\in H_{0}(z)$.

In this situation, in the definition (4) of the measure $\nu$, we set $|u_{j}(z)|:=\|u_{j}(z)\|_{H(z)}$, and the same for $|v_{j}(z)|$.

If $H_{0}(z)\equiv L$ $\mu$-a.e., where $\dim L=1$, then $\mathcal{H}_{0}$ can be identified with $L^{2}(\mu)$, and $x\odot y$ is just the pointwise product of $L^{2}$ functions $x$ and $y$.

We will use the formal columns $u=(u_{1},\dots,u_{m})^{t}$ and $v=(v_{1},\dots,v_{m})^{t}$. Then the perturbation in (1) can be expressed as follows:

[TABLE]

In what follows, instead of condition (D) we impose for convenience the following two conditions, that are formally stronger:

1. (D*′*)

The measure $\mu$ is dissectible; 2. (B)

The functions $u_{j}$ and $v_{j}$ are bounded.

If (K) holds, there is no loss of generality in assuming both (D*′*) and (B): one can achieve it by substituting $\mu$ with an equivalent measure. Indeed, the function $\rho=\sum_{j}|u_{j}|^{2}+|v_{j}|^{2}\in L^{1}(\mu)$ is positive and, by (K), nonzero $\mu$-a.e. Then the data

[TABLE]

satisfy (D*′*) and (B) and define new operators $N$ and $T$, which are unitarily equivalent to the original ones.

The construction of our functional model does not require condition (D*′*). However, in order to achieve our spectral decomposition goals the dissectability of the spectral measure is necessary.

Lemma 3.1**.**

Let $\Omega$ be a Jordan domain with admissible boundary with respect to $\mu$. Formula

[TABLE]

defines bounded operators defined on $L^{2}(\mu|\Omega)$ and with values on $E^{2}(\Omega^{c})$ and similarly from $L^{2}(\mu|\Omega^{c})$ to $E^{2}(\Omega)$.

Proof.

Let $f\in L^{2}(\mu)$ and assume that the closed support of $f$ does not touch $\partial\Omega$. For any function $g\in E^{2}(\Omega)$, analytic on a neighbourhood of the closure of $\Omega$, one has

[TABLE]

Since $\Omega$ is a Smirnov domain [13], functions $g$ as above form a dense subspace of $E^{2}(\Omega)$.

Notice also that $E^{2}(\Omega)$ is dual to $E^{2}_{0}(\Omega^{c})$ with respect to Cauchy duality (defined by the left hand side of the above identity). Accordingly, the boundedness of Carleson’s embedding operator  (9) implies the boundedness of the linear transform $f\mapsto C(f\mu)$, defined on $L^{2}(\chi_{\Omega}\cdot\mu)$ and with values in $E^{2}_{0}(\Omega^{c})$. The boundedness of the second operator is checked similarly. ∎

Definition 3.2**.**

To a domain $\Omega$ in $\mathbb{C}$ with admissible boundary we associate the following model space

[TABLE]

where $x\odot\bar{v}=(x\odot\bar{v}_{1},\dots,x\odot\bar{v}_{m})^{t}$ is a column function. According to (D*′*), $x\odot\bar{v}\in L^{2}(\mu)\otimes\mathbb{C}^{m}$.

This space is contained in $L^{1}(\Omega,dA)\otimes\mathbb{C}^{m}$, where $dA$ is the area measure. It follows from (K) that the function $x$ is determined by $g$ uniquely from $x\odot\bar{v}\cdot\mu=-\pi\bar{\partial}g$ on $\Omega$ (in the sense of distributions). Hence $b$ is also determined by $g$.

Proposition 3.3**.**

Let $\Omega$ be an admissible domain and $\gamma$ an admissible Jordan curve, which is either contained in $\Omega$ or is a subarc of $\partial\Omega$.

Then any function $g$ in $\operatorname{Mod}(\Omega)$ has a well-defined “trace” $g|\gamma$ on $\gamma$, which is an element of $L^{2}(\gamma)\otimes\mathbb{C}^{m}$. The map $g\mapsto g|\gamma\in L^{2}(\gamma)\otimes\mathbb{C}^{m}$ is bounded.

The exact definition of these traces will be given in Section 5.

Formula

[TABLE]

defines a Hilbert space structure on $\operatorname{Mod}(\Omega)$ (we leave the details to the reader). Proposition 3.3 implies that the maps

[TABLE]

are bounded on $\operatorname{Mod}(\Omega)$. Hence formula

[TABLE]

defines an equivalent Hilbert space norm on $\operatorname{Mod}(\Omega)$.

The $m\times m$ matrix function $\Psi$, defined by

[TABLE]

is called the perturbation matrix and will play a major role in the sequel. It belongs to $L^{2}_{loc}(\mathbb{C},dA)$. By Lemma 2.3, this function is defined and continuous on $\widehat{\mathbb{C}}\setminus\mathcal{G}_{s}(\mu)$, for any $s>0$.

Notice that $\Psi$ might be not continuous on the complement of the union of the sets $\mathcal{G}_{s}(\mu)$.

Put

[TABLE]

this scalar valued function is called the perturbation determinant. We observe that $\Psi$ and $\psi$ are holomorphic on $\widehat{\mathbb{C}}\setminus\operatorname{supp}\mu$, $\Psi(\infty)=I_{m}$ and $\psi(\infty)=1$.

For a domain $\Omega$ as in the above Definition, set

[TABLE]

so that

[TABLE]

A direct calculation yields the intertwining property

[TABLE]

We denote by $\operatorname{Der}\operatorname{supp}\mu$ the derivative set of the support of $\mu$, that is, the set of all accumulation points of $\operatorname{supp}\mu$.

Proposition 3.4**.**

Set $T_{0}=T|\mathcal{H}_{0}$.

1. (1)

The essential spectrum of $T_{0}$ coincides with the set $\operatorname{Der}\operatorname{supp}\mu$. 2. (2)

Suppose $\lambda\in\mathbb{C}\setminus\operatorname{supp}\mu$. Then $\lambda\in\sigma(T_{0})$ iff $\psi(\lambda)=0$. The same criterion guarantees $\lambda$ to belong to the point spectrum of $T_{0}$.

Proposition 3.5**.**

For any admissible domain $\Omega$ such that $\psi\neq 0$ on $\partial\Omega$, the linear manifold

[TABLE]

is a closed subspace of $\operatorname{Mod}(\Omega)$. Moreover, for any $a\in E^{2}(\Omega)\otimes\mathbb{C}^{m}$, the trace of $\Psi a$ on $\partial\Omega$, in the sense of Proposition 3.3, equals $\Psi\cdot a|\partial\Omega$ (notice that $\Psi$ is continuous on $\partial\Omega$).

Lemma 3.6**.**

Suppose $\Omega$ is a domain with admissible boundary. Then the operator

[TABLE]

is bounded on $\operatorname{Mod}(\Omega)$. Its spectrum is contained in $\overline{\Omega}$.

For any domain $\Omega$ as in the above Proposition 3.5, we consider the quotient space

[TABLE]

and call it the quotient model space corresponding to $\Omega$. The quotient multiplication operator  $M_{\Omega}$, induced by the above mapping (18), is correctly defined and bounded on $\mathcal{Q}(\Omega)$. We state the following analogue of Theorem 1 in [49].

Theorem 3.7**.**

Assume conditions (B) and (K) hold true. Let $\mathcal{B}$ be a domain with admissible boundary, such that $\sigma(T)\subset\mathcal{B}$. Then the transform

[TABLE]

is an isomorphism. It intertwines the operator $T_{0}$ with the quotient multiplication operator  $M_{\mathcal{B}}$.

This theorem shows that the perturbation matrix $\Psi$ is a close analogue of the characteristic function of a contractive or dissipative linear operator. This function appeared first in 1946 in the paper by Livsič [31], dedicated to quasi-hermitian operators with defect indices $(1,1)$.

The following two statements are analogous to [49], Lemmas 2 and 6.

Lemma 3.8** (Glueing lemma).**

Suppose $\Omega_{1}$, $\Omega_{2}$ and $\Omega$ are admissible domains, $\overline{\Omega}=\overline{\Omega}_{1}\cup\overline{\Omega}_{2}$, and $\overline{\Omega}_{1}\cap\overline{\Omega}_{2}$ is an arc. If $w_{j}\in\operatorname{Mod}(\Omega_{j})$ and $w_{1}=w_{2}$ on $\overline{\Omega}_{1}\cap\overline{\Omega}_{2}$ (in the sense of Proposition 3.3), then $w_{j}=w|\overline{\Omega}_{j}$ for a function $w\in\operatorname{Mod}(\Omega)$.

Theorem 3.9**.**

Assume conditions (B) and (K) are satisfied. Suppose $\mathcal{B}$ is an admissible domain which contains $\sigma(T)$, and let $\overline{\mathcal{B}}=\cup_{j=1}^{N}\overline{\Omega}_{j}$ be its finite partition, where $\Omega_{j}$ are open and disjoint admissible domains. Assume that the union of boundaries $\partial\Omega_{j}$ does not intersect $\mathcal{G}_{s}(\mu)$ for some $s>0$ and that the perturbation determinant $\psi$ does not vanish on $\cup_{j=1}^{N}\,\partial\Omega_{j}$.

Then $T$ splits into a direct sum

[TABLE]

where for each $j$, the spectrum of $T_{j}$ is contained in the closure of $\Omega_{j}$.

Formally, the above theorem does not require the dissectability condition (D*′*). Its main application, however, is for measures $\mu$ meeting this condition. In this case, the lengths of the sets $\operatorname{Re}\mathcal{G}_{s}(\mu)$ and $\operatorname{Im}\mathcal{G}_{s}(\mu)$ tend to zero as $s\to 0$. Therefore, by Lemmas  2.3 and  2.7, Theorem 3.9 permits one to dissect the spectrum along straight lines that cover densely the whole plane. This property is reflected in the abstract notion of a dissectible Banach space operator, which we introduce in Section 7. Theorem 7.2, which is a consequence of the above Theorem 3.9, gives a sufficient condition for a finite rank perturbation of a normal operator to be dissectible.

4. PROOFS OF LEMMAS FROM GEOMETRIC FUNCTION THEORY

In the present section we prove Lemmas 2.3,  2.7 and Proposition 2.9.

Lemma 4.1**.**

For any real number $x$, there is a strictly increasing sequence $\{x_{k}^{-}:k\geq 0\}$ of numbers of the form $x_{k}^{-}=m_{k}/2^{k}$, $m_{k}\in\mathbb{Z}$, such that

1) $\lim_{k}x_{k}^{-}=x$;

2) $2^{-k}\leq x-x_{k}^{-}\leq 2^{-k+1}$.

Proof.

Set $s_{k}$ to be the truncation of the binary representation of $x$ with $k$ digits after the point. Then $\{s_{k}\}$ increases, tends to $x$, and $0\leq x-s_{k}\leq 2^{-k}$. Now put $x_{k}^{-}=s_{k}-2^{-k}$. ∎

Proof of Lemma 2.3.

We can assume that $s=2^{-k_{0}}$ for some $k_{0}\geq 1$. Take any $f\in L^{\infty}(\nu)$ with $\|f\|_{L^{\infty}(\nu)}\leq 1$ and set $\eta=\eta_{f}=C(f\nu)$. Put $z=x+iy$, and let $\{x_{k}^{-}\}_{k\geq 1}$ be the sequence approximating $x$ from below, constructed in Lemma 4.1 and $\{x_{k}^{+}\}$ the sequence approximating $x$ from above, with analogous properties. Similarly, take two sequences $\{y_{k}^{-}\},\{y_{k}^{+}\}$, approximating $y$ from below and from above with the same properties. We have $x_{k}^{-}<x<x_{k}^{+}$, $2^{-k}\leq|x-x_{k}^{\pm}|\leq 2^{-k+1}$, and the same for $y$ and $\{y_{k}^{\pm}\}$. Consider the nested sequence of rectangles

[TABLE]

whose intersection is the point $z$. For any $k\geq k_{0}$, $R_{k-1}(z)\setminus R_{k}(z)$ is a union of no more than 16 dyadic squares from the generation $\mathcal{D}_{k}$. The (dyadic) squares $Q$ satisfying

[TABLE]

are disjoint, and their union over all $k\geq k_{0}$ equals to $R_{k_{0}-1}(z)\setminus\{z\}$. We arrange them in a single sequence $\{Q_{m}\}_{m\geq 1}$.

Notice that any square in the sequence $\{Q_{m}\}$ is at moderate distance from $z$ in the sense that

[TABLE]

Since $z\notin\mathcal{G}_{s}(\nu)$ and $|Q_{m}|\leq 2^{-k_{0}}$ for all $m\geq 1$, it follows that none of the squares $Q_{m}$ is heavy (see (3)). We also observe that $\nu(\{z\})=0$ (otherwise any sufficiently small dyadic square containing $z$ would be heavy, which would imply that $z\in\mathcal{G}_{s}(\nu)$). Whence

[TABLE]

where $g(\tau)=\int_{0}^{\tau}\frac{h(t)}{t^{2}}\,dt<\infty$. This proves the estimate (6).

With a small extra effort, we also derive the continuity of $\eta_{f}$ on $\mathbb{C}\setminus\mathcal{G}_{s}(\nu)$. Fix some $z\in\mathbb{C}\setminus\mathcal{G}_{s}(\nu)$, and let us prove the continuity of $\eta_{f}|\mathbb{C}\setminus\mathcal{G}_{s}(\nu)$ at $z$. It suffices to consider the case when $z\in(\operatorname{supp}\nu)\setminus\mathcal{G}_{s}(\nu)$.

First we isolate the following observation. Take some $w\in B(z,r)\setminus\mathcal{G}_{s}(\nu)$, where the radius $r$ is small. Choose $k_{1}$ so that $2^{-k_{1}-2}\leq r<2^{-k_{1}-1}$, and repeat the above construction of squares $Q_{m}$ and the estimate (20), taking $k_{1}$ in place of $k_{0}$. We infer that, for any $f$ as above and any $\varepsilon>0$, there exists $r>0$ such that

[TABLE]

whenever $w,z\in B(z,r)\setminus\mathcal{G}_{s}(\nu)$.

We check the continuity at $z$ directly by definition. Fix $\varepsilon>0$. Let $w\in\mathbb{C}\setminus\mathcal{G}_{s}(\nu)$, $|z-w|<\delta$, where $\delta$ is to be determined. Choose $r$ as above, and assume $\delta<r$. Then

[TABLE]

Since the last integral is continuous as a function of $w$ on the open disk $B(z,r)$, it follows that $|\eta_{f}(z)-\eta_{f}(w)|<\varepsilon$ whenever $w\in\mathbb{C}\setminus\mathcal{G}_{s}(\nu)$, $|z-w|<\delta$ and $\delta$ is sufficiently small. That is, $\eta_{f}$ is continuous on $\mathbb{C}\setminus\mathcal{G}_{s}(\nu)$. ∎

Next we pass to the proof of Lemma 2.7. We recall that a point $x_{0}$ is called a (Lebesgue) density point of a measurable set $A\subset\mathbb{R}$ if $x_{0}\in A$ and \mathcal{H}^{1}\big{(}[x_{0}-\varepsilon,x_{0}+\varepsilon]\cap A\big{)}/(2\varepsilon)\to 1 as $\varepsilon\to 0^{+}$. In virtue of Lebesgue Density Theorem, given a subset $A$ of $\mathbb{R}$ of positive measure, almost every point of $A$ is a density point.

Proof of Lemma 2.7.

It is easy to see that any scale function $h$ satisfies $h(t)\leq Ct$ for $t\in[0,1]$. Hence the Carleson constant of every set $\mathbb{C}\setminus\mathcal{G}_{s}(\mu)$ with respect to $\mu$ is bounded. Indeed, choose $k\geq 0$ with $2^{-k}<s$. Let $z\in\mathbb{C}\setminus\mathcal{G}_{s}(\mu)$. The closest point $p$ to $z$ in the discrete grid $2^{-k}\mathbb{Z}\times 2^{-k}\mathbb{Z}$ is the vertex of four dyadic squares from the generation $\mathcal{D}_{k}$. If $Q$ is any of these squares, then $|Q|\leq s$ and $10Q^{o}$ contains $z$, which implies that $\mu(Q)\leq h(|Q|)\leq C|Q|$. This shows that the constant $\operatorname{Carleson}(\mathbb{C}\setminus\mathcal{G}_{s}(\mu),\mu)$ is bounded for any $s$. Denote by $X_{s}$ and $Y_{s}$ the $x$- and $y$- projections of the set $\mathcal{G}_{s}(\mu)$. They are open and their lengths tend to zero as $s\to 0$. Note also that $\{X_{s}\}$ and $\{Y_{s}\}$ are increasing families of sets. Define $\mathbb{A}_{x}$ to be the set of points in $\mathbb{R}$ that are density points of at least one of the sets $\mathbb{R}\setminus X_{s}$. Define $\mathbb{A}_{y}$ similarly, using $Y_{s}$ in place of $X_{s}$. Then $\mathbb{A}_{x}$ and $\mathbb{A}_{y}$ are measurable, and their complements in $\mathbb{R}$ have Lebesgue measure zero. For any point $x_{0}$ in $\mathbb{A}_{x}$, there is some $s>0$ such that $x_{0}\in\mathbb{R}\setminus X_{s}$ and $x_{0}$ can be approximated both from above and from below by a sequence of points in the same set $\mathbb{R}\setminus X_{s}$. The points of $\mathbb{A}_{y}$ have a similar property.

This proves assertion 2).

Any broken line $\gamma$ as in Lemma is contained in $\mathbb{C}\setminus\mathcal{G}_{s}(\mu)$ for some sufficiently small $s$. In conclusion assertion 1) holds. ∎

Proof of Proposition 2.9.

*Case 1: * Assume first that the hypotheses hold, and all scale functions coincide: $h_{k}=h$ for all $k$. We normalize the measures $\mu_{k}$, assuming $\mu_{k}(\mathbb{C})=1$ for all $k$. We prove that there exist constants $c_{k}>0$ such that $\sum_{k}c_{k}\mu_{k}$ is a finite and $h$-dissectible measure. Put $c_{k}=\alpha_{k}^{2}$, where $\alpha_{k}>0$.

First choose positive numbers $s_{k}\to 0$ such that

[TABLE]

Note that if a square $Q$ is $\alpha_{k}\mu_{k}$-heavy, then

[TABLE]

Choose $\alpha_{k}\in(0,1)$ so that these inequalities imply that $|Q|\leq s_{k}$. We also assume

[TABLE]

With this choice, the measure $\nu:=\sum_{k}c_{k}\mu_{k}$ turns out to be finite and $h$-dissectible. Indeed, fix any $\varepsilon>0$. Find some $N$ such that $2^{-N}<\varepsilon/2$. Since each $\mu_{k}$ is $h$-dissectible, we can choose $q>0$ so that {\mathcal{H}}^{1}_{\rm cont}\big{(}\mathcal{G}_{q}(\mu_{k})\big{)}<\varepsilon/(2N) for $k=1,2,\dots,N$. We assert that whenever $0<s<q$,

[TABLE]

In view of (22) we find:

[TABLE]

and therefore, $\nu$ is dissectible, because $\varepsilon$ was arbitrary.

So it remains to check (23). In order to do so, take any dyadic square $Q$, which is $\nu$-heavy and satisfies $|Q|<s<q$. Then $\gamma\nu(Q)\geq\gamma h(|Q|)$, which we rewrite as

[TABLE]

Hence for some $k$,

[TABLE]

So $Q$ is $\alpha_{k}\mu_{k}$-heavy (and $\mu_{k}$-heavy). By the choice of $\alpha_{k}$, this implies $|Q|\leq s_{k}$. Hence

[TABLE]

where $t=\min(s,s_{k})<q$. This implies relation  (23).

*Case 2 (the general case): * Notice first that whenever $h$, $h_{1}$ are scale functions with $h_{1}\leq h$, then $Q$ $h$-heavy implies that $Q$ is $h_{1}$-heavy. Hence, if a measure $\nu$ is $h_{1}$-dissectible, then it is also $h$-dissectible with respect to any scale function $h$ such that $h_{1}\leq h$ on an interval $[0,\varepsilon]$, $\varepsilon>0$.

As a consequence, we will infer the general case from the above Case 1 once we check the following claim: *For any sequence of scale functions $\{h_{k}:k\geq 1\}$, there exists a scale function $h$ such that for any $k$, $h\geq h_{k}$ on some (nonempty) interval $(0,\varepsilon_{k}]$. *

This is simple. Namely, let $\{\beta_{k}\}$ be a sequence of positive numbers such that $\sum_{k}\beta_{k}<\infty$. Put

[TABLE]

where $g_{k}(t)=\min(h_{k}(t),\beta_{k})$. Then $h(t)$ is finite for any $t\geq 0$. If the numbers $\beta_{k}>0$ tend sufficiently rapidly to zero, then $h$ satisfies the integral condition (2) and so it is a desired scale function. ∎

5. PROOFS PERTAINING TO THE FUNCTIONAL MODEL

First we mention Cauchy-Pompeiu formula for functions belonging to the space $\operatorname{Mod}(\Omega)$: * If $\Omega$ is admissible and $w\in\operatorname{Mod}(\Omega)$, then*

[TABLE]

*(notice that $w|\partial\Omega\in L^{2}(\partial\Omega,|dz|)$). *

Indeed, by the definition of $\operatorname{Mod}(\Omega)$, it suffices to check this formula for functions $w=b\in E^{2}(\Omega)\otimes\mathbb{C}^{m}$ and for functions of the form $w=C(x\odot\bar{v}\cdot\mu)$, where $v\in L^{2}(\mu,H_{0})$. In the first case, it is just Cauchy’s formula. The second case is also obvious because $\bar{\partial}w=-\pi x\odot\bar{v}\cdot\mu$.

Proof of Proposition 3.3.

Let $g=C(x\odot\bar{v}\cdot\mu)+b$ be a function in $\operatorname{Mod}(\Omega)$, as in the Definition 3.2 of the model space and let $\gamma$ be a curve meeting the hypotheses. Suppose first that $\gamma\subset\Omega$.

If $x$ is bounded, then by (D*′*) and Lemma 2.3, one can define $g$ on the set $\Omega\setminus\cap_{s}\mathcal{G}_{s}(\mu)$; it will be continuous on $\Omega\setminus\mathcal{G}_{s}(\mu)$ for any $s>0$. Since $\gamma$ does not touch some set $\mathcal{G}_{s_{0}}(\mu)$, this permits one to define $g|\gamma$ as a restriction.

Now Lemma 3.1 implies that the map $g\to g|\gamma$ extends by continuity to functions $g\in\operatorname{Mod}(\Omega)$ as above, corresponding to arbitrary elements $x\in L^{2}(\mu,H_{0})$.

We adopt this extension as definition of the trace.

By the same Lemma 3.1, the map $g\to g|\gamma$ is also well-defined and bounded if $\gamma$ is a subarc of $\partial\Omega$. ∎

Proof of Proposition 3.4.

(1) The definition of $\mathcal{H}_{0}$ implies that the essential spectrum $\sigma_{ess}(N|\mathcal{H}_{0})$, and hence also $\sigma_{ess}(T_{0})$, coincide with $\operatorname{Der}\operatorname{supp}\mu$.

(2) Fix a point $\lambda$, which is not in the support of $\mu$. By the above, $\lambda\in\sigma(T_{0})$ if and only if $\lambda$ is an eigenvalue of $T_{0}$. This rewrites as $(z-\lambda)f(z)=-u^{t}c$, where $f\in\mathcal{H}_{0}$ ($f\neq 0$) is the corresponding eigenfunction and $c=\langle f,v\rangle$. It is easy to see that these two equations are solvable if and only if $\Psi(\lambda)c=0$ has a nonzero solution. ∎

Proof of Proposition 3.5.

Suppose $\partial\Omega$ admissible.

(a) First we check that \Psi\big{(}E^{2}(\Omega)\otimes\mathbb{C}^{m}\big{)} is contained in $\operatorname{Mod}(\Omega)$. Take any $a\in E^{2}(\Omega)\otimes\mathbb{C}^{m}$. We need to verify that $g:=\Psi a\in\operatorname{Mod}(\Omega)$. Note the identity

[TABLE]

Since Carleson’s embedding operator $J_{\Omega}$ is bounded on $E^{2}(\Omega)$ and $u$ is a bounded function, $x:=-(u^{t}\cdot a)\chi_{\Omega}$ belongs to $L^{2}(\mu)$. By (25), the function

[TABLE]

is analytic in $\Omega$. Let $\Omega_{n}$ be the domains that correspond to $\Omega$, whose existence is asserted in the Definition 2.5 of an admissible Jordan curve. Since Carleson’s constants of $\partial\Omega_{n}$ with respect to $\mu$ are uniformly bounded and $\Psi$ is bounded on the union of these curves (see Lemma 2.3), we deduce

[TABLE]

Therefore the integrals $\int_{\partial\Omega_{n}}|b|^{2}\,|dz|$ are uniformly bounded, which means that $b$ belongs to $E^{2}(\Omega)$. Hence $g=b+C(\bar{v}\odot x\cdot\mu)$ is in $\operatorname{Mod}(\Omega)$.

The same arguments imply the boundedness of the map $a\to\Psi a$, acting on $E^{2}(\Omega)\otimes\mathbb{C}^{m}$ and with values on $\operatorname{Mod}(\Omega)$.

(b) We verify now the last statement referring to the trace of $\Psi a$ on $\partial\Omega$. Let $C_{a}({\overline{\Omega}})$ stand for the space of functions, analytic in $\Omega$ which can be continuously extended to the boundary. Consider first the case $a\in C_{a}({\overline{\Omega}})\otimes\mathbb{C}^{m}$ and define $x,b$ as above. Then $x\in L^{\infty}(\mu)$, which implies that $C(\bar{v}\odot x\cdot\mu)$ is well-defined at any point of $\partial\Omega$ and is continuous on this curve. Hence $b$ is also continuous on $\partial\Omega$ and so $b\in C_{a}({\overline{\Omega}})\otimes\mathbb{C}^{m}$. Thus equality $\Psi a=C(\bar{v}\odot x\cdot\mu)+b$ holds pointwisely on $\partial\Omega$, which implies that the trace of $\Psi a$ on $\partial\Omega$ equals $\Psi\cdot a|\partial\Omega$.

The case of a general element $a$ in $E^{2}(\Omega)\otimes\mathbb{C}^{m}$ is obtained by approximating $a$ in the norm of $E^{2}(\Omega)\otimes\mathbb{C}^{m}$ by a sequence of functions $a_{n}$ in $C_{a}({\overline{\Omega}})\otimes\mathbb{C}^{m}$ and by invoking the boundedness of the trace map $w\mapsto w|\partial\Omega$, $w\in\operatorname{Mod}(\Omega)$.

(c) To prove that \Psi\big{(}E^{2}(\Omega)\otimes\mathbb{C}^{m}\big{)} is a closed subspace of $\operatorname{Mod}(\Omega)$, we use expression (15), which gives an equivalent norm on $\operatorname{Mod}(\Omega)$. Since $\|\Psi^{-1}\|$ is uniformly bounded on $\partial\Omega$, it follows that $\|\Psi a\|_{\operatorname{Mod}(\Omega)}\geq\delta\|a\|_{E^{2}}$ for any $a\in E^{2}(\Omega)\otimes\mathbb{C}^{m}$, where $\delta>0$. Hence the image of the map $a\in E^{2}(\Omega)\otimes\mathbb{C}^{m}\mapsto\Psi a$ is closed in $\operatorname{Mod}(\Omega)$. ∎

Proof of Lemma 3.6.

Let $g\mapsto x(g)$ be the map defined in (14). Since $x(g)$ is obtained via the Cauchy-Riemann operator $\bar{\partial}g$, it follows that $x(ag)=ax(g)$ for $g\in\operatorname{Mod}(\Omega)$ and $a$ analytic in a neighborhood of $\overline{\Omega}$. Applying this observation to $a(z)=z$ and using the equivalent norm (15) on the model space, we deduce the boundedness of the operator $g\mapsto zg$ on $\operatorname{Mod}(\Omega)$. For the same reasons, for any $\lambda\in\Omega^{c}$, the multiplication operator $g\mapsto(z-\lambda)^{-1}g$ provides a bounded inverse to the operator $g\mapsto(z-\lambda)g$. ∎

Proof of Theorem 3.7.

We can repeat the arguments in the proof of Theorem 1 in [49] (see [49], p. 66). The intertwining property follows from  (17). It remains to check that, given $w_{0}$ in $\operatorname{Mod}_{B}$, there exist a unique $x$ in $L^{2}(\mu,H_{0})$ and a unique $a\in E^{2}(\mathcal{B})\otimes\mathbb{C}^{m}$ satisfying

[TABLE]

We appeal to a class of Toeplitz type operators. If $F$ is matrix-valued function in $L^{\infty}(\partial\mathcal{B},L(\mathbb{C}^{m}))$, then the Toeplitz operator $\tau_{F}$ acts on $E^{2}(\mathcal{B})\otimes\mathbb{C}^{m}$ by

[TABLE]

It is bounded. It is easy to verify that $\tau_{G}\tau_{F}=\tau_{GF}$ whenever $F,G\in H^{\infty}(\mathcal{B}^{c},L(\mathbb{C}^{m}))$. In particular, $\tau_{\Psi}$ is invertible, and $\tau_{\Psi}^{-1}=\tau_{\Psi^{-1}}$.

Notice that, for a function $g$ in $\operatorname{Mod}(\mathcal{B})$, one has

[TABLE]

Therefore (26) is solvable with respect to $x$ if and only if

[TABLE]

If we define $a$ by this formula, by Proposition 3.5, $\Psi a$ and hence $w:=w_{0}+\Psi a$ belong to $\operatorname{Mod}(\mathcal{B})$. Therefore $a$ and $x$ are determined by (26). ∎

Proof of the Glueing lemma 3.8.

Suppose $w_{1}$, $w_{2}$ satisfy the hypotheses and define a function $w$ on $\Omega_{1}\cup\Omega_{2}$ by w\big{|}_{\Omega_{j}}=w_{j}. By summing formulas (24) applied to $\Omega_{1}$, $\Omega_{2}$, we infer

[TABLE]

on $\Omega_{1}\cup\Omega_{2}$. The right hand part is a function in $\operatorname{Mod}_{\Omega}$. ∎

We remark that the same lemma holds for finitely many bordering domains, instead of two of them.

Proof of Theorem 3.9.

We can follow the lines of the proof of Lemma 6 in [49]. First we prove the assertion for the operator $T_{0}=T|\mathcal{H}_{0}$, whose functional model is given by Theorem 3.7. Consider the restriction maps $\mathcal{J}_{k}w=w|\Omega_{k}\in\operatorname{Mod}(\Omega_{k})$, $w\in\operatorname{Mod}(\Omega)$. It follows from Proposition 3.3 and the expression (15) for the norm in the model spaces that these are bounded linear operators. They induce operators $\widehat{\mathcal{J}}_{k}:\mathcal{Q}(\Omega)\to\mathcal{Q}(\Omega_{k})$, which are well-defined and bounded. We assert that $\widehat{\mathcal{J}}=(\widehat{\mathcal{J}}_{1},\dots,\widehat{\mathcal{J}}_{N}):\mathcal{Q}(\Omega)\to\oplus_{k}\mathcal{Q}(\Omega_{k})$ is an isomorphism. Let $w\in\operatorname{Mod}(\Omega)$, and let $\hat{w}$ be the corresponding element (class of equivalence) in $\mathcal{Q}(\Omega)$. If $\widehat{\mathcal{J}}\hat{w}=0$, then $w|\Omega_{k}=\Psi a_{k}$ for some functions $a_{k}\in E^{2}(\Omega_{k})$, $k=1,\dots,N$. It is easy to check that for any $g\in\operatorname{Mod}(\Omega)$ and any domains $\Omega_{j}$, $\Omega_{k}$, bordering by an arc $\gamma$, the trace of $g|\Omega_{j}$ on $\gamma$ equals the trace of $g$ on $\gamma$. Hence, by the last statement of Proposition 3.5, $a_{k}=a_{j}$ on $\partial\Omega_{k}\cap\partial\Omega_{j}$, $1\leq k<j\leq N$. By applying the Cauchy integral representation of $E^{2}$ functions, one finds there is some $a\in E^{2}(\Omega)\otimes\mathbb{C}^{m}$ such that $a_{k}=a|_{\Omega_{k}}$ for all $k$. This shows that $\ker\widehat{\mathcal{J}}=0$.

The fact that $\widehat{\mathcal{J}}$ is also onto is shown in the same way as in the proof of Lemma 6 in [49], and we leave the details to the reader.

In the general case, the assertion follows, because $T|\mathcal{H}\ominus\mathcal{H}_{0}$ is normal and $T|\mathcal{H}_{0}$ has the same perturbation determinant as $T$. ∎

6. THE EXISTENCE OF INVARIANT SUBSPACES

The aim of this section is to prove the following result.

Theorem 6.1**.**

Suppose the linear operator $T$ is given by (1) and satisfies conditions (D) and (K). If either $\operatorname{Der}\operatorname{supp}\mu$ is not connected or there exists a domain $G$, whose boundary is a Lipschitz Jordan curve, such that $G\cap\operatorname{Der}\operatorname{supp}\mu=\emptyset$ and the intersection of $\operatorname{supp}\mu$ with the boundary of $G$ contains an arc, then $T$ has a nontrivial invariant subspace.

The proof of this result will rely on the following known fact.

Theorem A** (Privalov’s uniqueness theorem, see [39], Ch. IV, §2.6).**

If a function $f(z)$, meromorphic on a domain, bounded by a rectifiable Jordan curve $\Gamma$, has angular limit values equal to zero on a subset $E$ of $\Gamma$ with $\mathcal{H}^{1}(E)>0$, then $f$ is identically zero.

Recall also F. and M. Riesz theorem [42, 40]: if $\Omega$ is any Jordan domain, bounded by a rectifiable curve, then the Hausdorff measure $\mathcal{H}^{1}$ and the harmonic measure are mutually absolutely continuous on $\partial\Omega$.

Proof of Theorem 6.1.

We can assume (D*′*), (B) and also that $\mathcal{H}=\mathcal{H}_{0}$. We start by the following reduction. Consider the open set

[TABLE]

Note that $\Psi$ and $\psi$ are meromorphic functions on $\Omega$. If $\psi$ vanishes on one of its connected components, then, by part (2) of Proposition 3.4, $T$ has an eigenvalue, and therefore has an invariant subspace. So, from now on, we will assume that $\psi$ does not vanish identically on any of the connected components of $\Omega$.

Case 1: The (compact) set $\operatorname{Der}\operatorname{supp}\mu$ is disconnected. Then it decomposes into a disjoint union of two non-empty closed and relatively open subsets, say, $F_{1}$ and $F_{2}$. Since the zeros and the poles of $\psi$ form discrete subsets of the complement of $F_{1}\cup F_{2}$, one can find open disjoint sets $\mathcal{O}_{1}\supset F_{1}$ and $\mathcal{O}_{2}\supset F_{2}$, whose boundaries are finite unions of rectifiable Jordan curves, do not intersect with the set $F_{1}\cup F_{2}$ and do not contain neither zeros nor poles of $\psi$. Note that $\partial\mathcal{O}_{1}$ and $\partial\mathcal{O}_{2}$ are contained in $\mathbb{C}\setminus\sigma(T)$, whereas $F_{1}$ and $F_{2}$ are contained in $\sigma(T)$. It follows that the corresponding Riesz projection $P_{\mathcal{O}_{1}}$ is a non-trivial idempotent, commuting with $T$. The range of $P_{\mathcal{O}_{1}}$ is a nontrivial invariant subspace of $T$.

Case 2: There exists a domain $G$, meeting the hypotheses of the theorem. By passing to a smaller domain, we may assume that $\operatorname{supp}\mu\cap\partial G$ has the form

[TABLE]

where $J\subset\mathbb{R}$ is a finite closed interval and $f$ is a Lipschitz function. Since $\psi$ is meromorphic in $G$ and is not identically zero in this domain, Privalov’s uniqueness theorem implies $\psi(\lambda)\neq 0$ for a.e. $\lambda\in\gamma$. We can also assume that $f$ is non-constant on $J$.

There is a compact subset $F$ of $f(J)\cap\mathbb{A}_{y}$ of positive measure. Since $f$ maps sets of zero measure to sets of zero measure and $f(f^{-1}(F))=F$, the preimage $f^{-1}(F)$ has positive measure. Hence one can choose $z_{0}=x_{0}+iy_{0}\in\gamma$ so that $x_{0}\in J^{o}\cap\mathbb{A}_{x}\cap f^{-1}(\mathbb{A}_{y})$ and $\psi(z_{0})\neq 0$. Then $y_{0}\in\mathbb{A}_{y}$.

Part 2) of Lemma 2.7 implies that there exist sequences $\{x_{n}^{-}\},\{x_{n}^{+}\}$ tending to $x_{0}$ and $\{y_{n}^{-}\},\{y_{n}^{+}\}$ tending to $y_{0}$ such that $x_{n}^{-}<x_{0}<x_{n}^{+}$ and $y_{n}^{-}<y_{0}<y_{n}^{+}$ for all $n$. Moreover, there is $s>0$ such that for all $n$, $x_{n}^{-},x_{n}^{+}\in\mathbb{R}\setminus\operatorname{Re}\mathcal{G}_{s}(\mu)$ and $y_{n}^{-},y_{n}^{+}\in\mathbb{R}\setminus\operatorname{Im}\mathcal{G}_{s}(\mu)$.

Consider a rectangle $R=[x_{n}^{-},x_{n}^{+}]\times[y_{n}^{-},y_{n}^{+}]$. If $n$ is sufficiently large, then $|\psi|>\delta>0$ on $R\setminus\mathcal{G}_{s}(\mu)$ and $\gamma$ is not contained in $R$.

Notice also that the boundary of $R$ is contained in $\mathbb{C}\setminus\mathcal{G}_{s}(\mu)$. Therefore $R^{o}$ is an admissible domain.

By Theorem 3.9, $T$ has invariant subspaces $L$ and $M$ such that $\mathcal{H}_{0}=L\dotplus M$, $\sigma(T|L)\subset\sigma(T)\cap R$ and $\sigma(T|M)\subset\sigma(T)\setminus R^{o}$. Notice that $\sigma(T)=\sigma(T|L)\cup\sigma(T|M)$. Therefore each one of the spectra $\sigma(T|L)$ and $\sigma(T|M)$ contains a nontrivial subarc of $\gamma$. Hence $L\neq 0$ and $M\neq 0$, so that $L$ is a nontrivial invariant subspace of $T$. ∎

7. BISHOP PROPERTIES ON THE MODEL SPACE AND DECOMPOSABILITY

A landmark contribution to axiomatic spectral theory is Bishop’s 1959 article [8]. Inspired by generalized spectral decompositions of linear and bounded Banach space operators lacking a spectral measure, he has identified four different behaviors of the resolvent of the dual which imply the existence of invariant subspaces localizing the spectrum. Soon afterwards Foiaş has isolated in 1963 the concept of decomposable operator [17]; this class of linear operators and Bishop’s properties have simplified and unified conceptually many lines of research in spectral analysis and produced over the years far reaching applications. For an early account of the theory we refer to [10, 12] as for recent developments, including multivariate generalizations, see [15].

A linear and bounded operator $T$ acting on a Banach space $X$ is called decomposable, if for every finite open cover of its spectrum

[TABLE]

there are $T$-invariant subspaces $X_{j}\subset X,\ \ 1\leq j\leq n,$ with the properties

[TABLE]

and

[TABLE]

Note that the above decomposition is not a direct sum, nor there are bounded linear projections onto its terms. Later on it was proved that only open covers with two sets suffice for the decomposability condition.

Examples are all classes of operators possessing a spectral measure, and beyond, for instance operators admitting a functional calculus with smooth functions.

A decomposable operator $T\in L(X)$ has the single valued extension property: for every open set $U\subset\mathbb{C}$ and any vector valued analytic function $f\in{\mathcal{O}}(U,X)$, $(zI-T)f(z)=0,\ \ z\in U,$ implies $f=0$. As soon as an operator $T\in L(X)$ has the single valued extension property, one can speak without ambiguity about the localized spectrum $\sigma_{x}(T)$ of $T$ with respect to a vector $x\in X$, defined as the smallest closed subset of the complex plane allowing the localized resolvent $(zI-T)^{-1}x$ to have an analytic continuation on its complement [10].

One step further, the operator $T\in L(X)$ satisfies Bishop’s property $(\beta)$ if the map

[TABLE]

is one to one with closed range for every open set $U\subset\mathbb{C}$. Here $\mathcal{O}(U,X)=\mathcal{O}(U)\otimes X$ stands for the Fréchet space of all analytic $X$-valued functions on $U$. Obviously it is sufficient to check this condition on open disks $U$.

A decomposable operator possesses property $(\beta)$ [8, 10]. A more recent theorem due to Albrecht and Eschmeier [2] completes Bishop’s visionary program by stating that $T$ is decomposable if and only if $T$ and its dual $T^{\ast}$ both have property $(\beta)$. All these results have an analog in the case of commuting tuples of operators. In that context the analytic sheaf model

[TABLE]

is prevalent, opening the gate to homological algebra techniques [15].

Prompted by the spectral behavior our main theorem reveals, we propose the following more restrictive variation of the decomposability property. If $A\subset B\subset\mathbb{C}$, we denote by $\partial(A;B)$ the relative boundary of $A$ in $B$.

Definition 7.1**.**

Let $X$ be a Banach space. We will say that an operator $T\in L(X)$ is dissectible if for any open cover

[TABLE]

there are closed sets $F_{j}\subset U_{j}$ such that $\sigma(T)=\cup_{j=1}^{n}F_{j}$, $F_{j}\cap F_{k}=\partial(F_{j};\sigma(T))\cap\partial(F_{k};\sigma(T))$ for all $j\neq k$, and there are $T$-invariant subspaces $X_{j}\subset X,\ \ 1\leq j\leq n,$ such that

[TABLE]

and a direct sum decomposition holds:

[TABLE]

It is clear that this notion is stronger than decomposability. Any normal operator and any compact operator are dissectible. On the other hand, plenty of decomposable operators are not dissectible.

To fix ideas we discuss a simple case. Let $Y\subset\mathbb{C}$ be any connected compact set, which has more than one point. Let $X=C(Y)$ be the space of continuous functions on $Y$. The multiplication operator $Tf(z)=zf(z)$ is decomposable but not dissectible.

Indeed, it is easy to see that $\sigma(T)=Y$. Take any open cover $Y\subset U_{1}\cup U_{2}$ of $\sigma(T)$ such that neither $U_{1}$ nor $U_{2}$ covers $Y$. Notice that if $F_{1}$, $F_{2}$ correspond to this open cover, then $f|\partial(F_{j};Y)=0$ for all $f$ in $X_{j}$. Since $Y=F_{1}\cup F_{2}$ is connected and $F_{j}\neq\emptyset$, there is a point $w$ in $F_{1}\cap F_{2}=\partial(F_{1};Y)\cup\partial(F_{2};Y)$. By (30), any function in $X$ vanishes at $w$, a contradiction.

To increase generality, take now $X$ to be a continuously embedded Banach space into $C(Y)$, so that $T=M_{z}$ on $X$ is decomposable and $\sigma(T)=Y$ (such as a Sobolev space). The argument above adapts and implies that $T$ is not dissectible.

We observe that Theorem 3.9 permits us to prove the following fact.

Theorem 7.2**.**

Assume that conditions (D) and (K) are satisfied. If, moreover, for any $s>0$, the one-dimensional Hausdorff measure of the set

[TABLE]

equals to zero, then $T$ is dissectible.

We recall that, by Lemma 2.3, $\psi$ is continuous on $\mathbb{C}\setminus\mathcal{G}_{s}(\mu)$ for any $s>0$; the sets $\mathcal{G}_{s}(\mu)$ have been defined in (3).

In the proof, we use the following notation. Given some $z=x+iy$ and some radius $r>0$, we set

[TABLE]

to be the left and the right half of the disc $B(z,r)$. These two sets are open.

Let $Y\subset\mathbb{C}$, and let $z\in Y$. We will say that $z$* is $r$-accessible from the right in $Y$* if the intersection $Y\cap B_{+}(z,r)$ is not empty.

We say that a point $z$ is accessible from the right in $Y$ if it $r$-accessible from the right in $Y$ for any positive $r$. Equivalently, there should exist a sequence $\{w_{k}\}$ of points of $Y$ that tends to $z$ and satisfies $\operatorname{Re}w_{k}>\operatorname{Re}z$ for all $k$.

Note that $z$ is inaccessible from the right whenever it not $r$-accessible from the right for some $r>0$. We define similarly the accessibility from the left, from above and from below.

Lemma 7.3**.**

For any bounded subset $Y$ of the complex plane, the set of its points, inaccessible from the right in $Y$, is contained in a countable union of vertical (straight) lines.

Proof.

A point of $Y$ is inaccessible from the right (in $Y$) if and only if it is $1/k$-inaccessible from the right for some $k\in\mathbb{N}$. So it will be enough to check that, say, for any $r\in(0,2)$, the set of points of $Y$, $r$-inaccessible from the right, can be covered by finitely many vertical lines.

Fix some $r\in(0,2)$ and assume that $Y\subset B(z_{0},R)$ for some $z_{0}$ and some $R>0$. We will prove that the above set of points can be covered by no more than $[4(R+1)^{2}/r^{2}]$ vertical lines, where $[t]$ stands for the entire part of $t$.

Suppose it is false. Then there exist $N>4(R+1)^{2}/r^{2}$ points $z_{j}\in Y$ that are $r$-inaccessible from the right and all have distinct real parts. By comparing areas, we get that for some $k\neq\ell$, the discs $B(z_{k},r/2)$ and $B(z_{\ell},r/2)$ have to intersect. Hence $|z_{k}-z_{\ell}|<r$. Let, for instance, $\operatorname{Re}z_{k}<\operatorname{Re}z_{\ell}$. Then $z_{\ell}\in B_{+}(z_{k},r)\cap Y$. Therefore $z_{k}$ is $r$-accessible from the right, a contradiction. This finishes the proof. ∎

Proof of Theorem 7.2.

By applying last Lemma four times to the set $Y=\sigma(T)$, we find that in Lemma 2.7 the sets $\mathbb{A}_{x},\mathbb{A}_{y}\subset\mathbb{R}$ can be chosen so that they satisfy the following additional properties:

1. (P1)

Any point $z\in\sigma(T)$ such that $\operatorname{Re}z\in\mathbb{A}_{x}$ is accessible from the right and from the left in $\sigma(T)$; 2. (P2)

Any point $z\in\sigma(T)$ such that $\operatorname{Im}z\in\mathbb{A}_{y}$ is accessible from above and from below in $\sigma(T)$. 3. (P3)

The lines $\{\operatorname{Re}z=x_{0}\}$, where $x_{0}\in\mathbb{A}_{x}$, and $\{\operatorname{Im}z=y_{0}\}$, where $y_{0}\in\mathbb{A}_{y}$, are contained in $\{z\in\mathcal{G}_{s}:\psi(z)\neq 0\}$ for some positive $s$.

We also assume that whenever $a$ is an isolated point of $\sigma(T)$, $\operatorname{Re}a\notin\mathbb{A}_{x}$ and $\operatorname{Im}a\notin\mathbb{A}_{y}$. This is achieved by quitting some countable subsets from $\mathbb{A}_{x}$ and $\mathbb{A}_{y}$, once again.

Suppose an open cover (28) of $\sigma(T)$ is given. Draw finitely many vertical lines $\{\operatorname{Re}z=x_{j}\}$ ($1\leq x_{j}\leq N$), where $x_{j}\in\mathbb{A}_{x}$, and finitely many horizontal lines $\{\operatorname{Im}z=y_{j}\}$ ($1\leq y_{j}\leq M$), where $y_{j}\in\mathbb{A}_{y}$. Let us assume that the finite sequences $\{x_{j}\}$, $\{y_{j}\}$ are increasing and that the open rectangle $(x_{1},x_{N})\times(y_{1},y_{M})$ contains $\sigma(T)$. Put $R_{jk}=[x_{j},x_{j+1}]\times[y_{k},y_{k+1}]$. By Lebesgue Lemma, we can also assume that the lines that were drawn are so close to each other that for each pair $(j,k)$, there is an index $\hat{m}(j,k)$ such that $R_{jk}\subset U_{\hat{m}(j,k)}$. Fix these numbers $\hat{m}(j,k)$, and set

[TABLE]

Then $\widetilde{R}_{m}$ have disjoint interiors and $\widetilde{R}_{m}\subset U_{m}$.

The desired sets $F_{m}$ will be defined as

[TABLE]

where $R_{m}$ are certain modifications of $\widetilde{R}_{m}$. These modifications are performed as follows. By a vertex, we mean a point of the form $(x_{j},y_{k})$, which is on the boundary of one of polygonal sets $\widetilde{R}_{m}$. We say that a vertex $p$ is special if it is a limit point of $\sigma(T)\cap\widetilde{R}_{m}$ only for one index $m=m(p)$. (For any vertex $p\in\sigma(T)$, such index $m$ should exist.)

For any special vertex $p$, choose a small closed rectangle $\rho=\rho(p)$, whose vertices lie on $\mathbb{A}_{x}+i\mathbb{A}_{y}$, that contains $p$ in its interior and does not touch any drawn line, except those that pass through $p$. We also require that $\rho\subset U_{m(p)}$ and that $\rho\cap\sigma(T)=\widetilde{R}_{m(p)}\cap\sigma(T)$.

Notice that $p$ cannot be a limit point of $\sigma(T)\cap\partial\widetilde{R}_{m}$ (otherwise, as it is easy to see from (P1) and (P2) above, $p$ would be also a limit point of $\sigma(T)\cap\widetilde{R}_{t}$ for some $t\neq m$). Therefore $\rho$ can be chosen so that

[TABLE]

We make replacements

[TABLE]

and

[TABLE]

whenever $t\neq m$ and $p$ lies on the boundary of $\widetilde{R}_{t}$. After making these replacements for all special vertex points $p$, we obtain modified sets $R_{m}$, such that special vertex points are no longer their vertices. (We may assume that the rectangles $\rho(p)$, corresponding to different special vertex points $p$, are disjoint, so that the result does not depend on the order of these replacements.) Observe that $R_{m}$ have disjoint interiors and $R_{m}\subset U_{m}$.

Define the sets $F_{m}$ by (31). By (32), all vertices of the sets $R_{m}$ that were not among the vertices of $\widetilde{R}_{m}$ do not belong to $\sigma(T)$.

By (P3) and Theorem 3.9, there is a decomposition $T=T_{1}\dotplus\dots\dotplus T_{N}$, where $\sigma(T_{m})\subset F_{m}$.

It remains to check the condition concerning the boundaries of $F_{m}$. Let $w\in F_{m}\cap F_{t}$, with $m\neq t$. Then $w$ belongs to the boundaries of $R_{m}$ and of $R_{t}$. If $w$ is not a vertex point, then it follows from (P1) and (P2) that $w$ is a limit of points in $\sigma(T)\setminus F_{m}$, and so $w\in\partial(F_{m};\sigma(T))$. Similarly, $w\in\partial(F_{t};\sigma(T))$.

Finally, let $w$ be a vertex point. Since $w\in\sigma(T)$, it is not a special point. Therefore there are indices $k\neq\ell$ such that $w$ is a limit point of both sets $F_{k}$ and $F_{\ell}$. Either $k\neq m$ or $\ell\neq m$; assume for instance that $k\neq m$. Then $w$ is a limit point of the set $\sigma(T)\cap R_{k}^{o}$, which does not intersect $F_{m}$, and therefore $w\in\partial(F_{m};\sigma(T))$. Similarly, $w\in\partial(F_{t};\sigma(T))$.

In conclusion, $F_{m}\cap F_{t}=\partial(F_{m};\sigma(T))\cap\partial(F_{t};\sigma(T))$. ∎

It turns out that a sufficient condition for decomposability of a totally different nature can be proved.

Indeed, define

[TABLE]

(the right hand part is understood as zero if the matrix $\Psi(z)$ is not invertible). Notice that for a rank one perturbation, when $m=1$, $\tilde{\psi}=\psi=\Psi$. We also remark that $|\tilde{\psi}(z)|\leq\|\Psi(z)\|$ for a.e. $z$, so that $\tilde{\psi}$ is a nonnegative locally $L^{1}$ function. The set of zeros of $\psi$ and of $\tilde{\psi}$ on each set $\mathbb{C}\setminus\mathcal{G}_{s}(\mu)$ coincide.

Definition 7.4**.**

We say that $\tilde{\psi}$ has no deep zeros if for any bounded domain $\mathcal{D}$ and any its compact subset $K$, there is a constant $C(\mathcal{D},K,\tilde{\psi})$ such that the estimate

[TABLE]

holds for any function $f$, holomorphic in $\mathcal{D}$.

Theorem 7.5**.**

If the function $\tilde{\psi}$ has no deep zeros, then $T$ has property $(\beta)$.

In virtue of a result proved by Domar [11], we derive a more tangible criterion.

First we define an auxiliary function $F^{*}$. Choose a disc $B(0,R)$, containing the spectrum of $T$, and let $F^{*}$ be the decreasing rearrangement of $\log\log(e+\tilde{\psi}^{-1})|B(0,R)$. That is, $F^{*}$ is a decreasing non-negative function on $[0,\pi R^{2}]$ such that, for any $s>0$, the length of the interval $\{t:F^{*}(t)\geq s\}$ is equal to the area measure of the set $\{z\in B(0,R):\log\log(e+\tilde{\psi}(z)^{-1})\geq s\}$.

Theorem 7.6**.**

As usual, assume (D′), (B) and (K). If $\Psi$ is a Hölder-$\alpha$ function for some $\alpha\in(0,1]$ and

[TABLE]

for some positive $\varepsilon$, then $T$ has property $(\beta)$.

Theorem 7.6 has a simple corollary concerning decomposability.

Corollary 7.7**.**

Suppose the conditions (D′) and (B). Suppose that for $\mu$-a.e. $z$, the linear span of the vectors $u_{1}(z),\dots,u_{m}(z)$ coincides with the linear span of the vectors $v_{1}(z),\dots,v_{m}(z)$. If $\Psi$ is a Hölder-$\alpha$ function for some $\alpha\in(0,1]$ and (34) holds, then $T$ is decomposable.

Indeed, one obtains the representation of $T^{*}$ as a perturbation of $N^{*}$ by passing to conjugates in (1). The hypotheses on $u_{j}$ and $v_{j}$ imply that both representations of $T$ and of $T^{*}$ satisfy condition (K). Moreover, in this case the corresponding perturbation matrices $\Psi_{T}$ and $\Psi_{T^{*}}$ are of size $m$ and satisfy $\Psi_{T^{*}}(z)=\Psi_{T}(\bar{z})^{*}$. (If only (K) is assumed, one has to add new “fake” terms to the representation of $T^{*}$, which makes the size of $\Psi_{T^{*}}$ greater than $m$.) Hence $\tilde{\psi}_{T^{*}}(z)=\tilde{\psi}_{T}(\bar{z})$. So we derive property ($\beta$) for both $T$ and $T^{*}$, and therefore $T$ is decomposable.

Example 7.8**.**

The decomposability can fail if $\psi$ is a smooth function that vanishes on an smooth arc $\delta$ and decays very rapidly when approaching to this arc. This was (rather briefly) explained in [49] at the end of Section 6. We reproduce the argument here with a few more details. Assume that $\mu$ is absolutely continuous with respect to area measure and $m=1$. Assume also that $\mathcal{H}=\mathcal{H}_{0}$. Notice the following simple fact: if $S\in L(X)$ is a Hilbert space operator and $W:K\to C(\delta)$ is its diagonalization, that is, $(WSx)(z)=z(Wx)(z)$ for all $x\in X$, then $(Wx)|\delta\setminus\sigma(S)=0$, $x\in X$. Indeed, it follows that the function $\lambda\mapsto(\cdot-\lambda)^{-1}Wx\in C(\delta)$ extends analytically from $\mathbb{C}\setminus(\sigma(S)\cup\delta)$ to $\mathbb{C}\setminus\sigma(S)$.

Fix a domain $\mathcal{B}\supset\sigma(T)$. If $\delta\subset\sigma(T)$ and $\psi$ and $\bar{\partial}\psi$ decay sufficiently fast when approaching $\delta$, then, as explained in [49], the operator $Wf:=f|\delta$ is a well-defined diagonalization operator from the model space $\mathcal{Q}(\mathcal{B})$ to a space of quasianalytic functions on $\delta$. Take now any open cover $\sigma(T)\subset U_{1}\cup U_{2}$ such that each of the sets $\delta\setminus U_{1}$ and $\delta\setminus U_{2}$ contains a subarc of $\delta$. If there exists the corresponding decomposition $L^{2}(\mu,H_{0})=X_{1}+X_{2}$, as in  (29), then $(Wx)|\delta\setminus U_{j}=0$ for $x\in X_{j}$. By quasianalyticity, $(Wx)|\delta=0$ for all $x\in X_{j}$ and hence for all $x\in L^{2}(\mu,H_{0})$, which is obviously false. This shows that $T$ cannot be decomposable.

Remark 7.9**.**

Notice also that Corollary 7.7 ensures decomposability in many cases when the zero set of $\psi$ is larger than it is allowed by Theorem 7.2 (which assumes it to be of zero length). The arguments similar to those in the above example show that, whenever the zero set of $\psi$ on some set $\mathbb{C}\setminus\mathcal{G}_{s}(\mu)$ is connected, operator $T$ will not be dissectible, in general.

However, Corollary 7.7 says nothing if the zero set of $\psi$ contains an open set $U$. If $T$ is a rank one perturbation of $N$, then by applying the model Theorem 3.7 and the diagonalization map $w\in\mathcal{Q}(\mathcal{B})\mapsto w|U$ (which is well-defined, because $\Psi|U=\psi|U=0$), it is easy to see that $T$ is not decomposable. We do not know whether this fact extends to higher rank perturbations.

Notice that condition (34) resembles the well-known decomposability criterion due to Lyubich and Macaev [32].

Diagonalization operators as above are certainly important in the spectral study of perturbations of normal operators and have been used intensively in [49]. Some additional material can be found in unpublished preprint [50] by the second author, where completeness of generalized eigenvectors of $T$ of several kinds has been discussed. Later, the local spectral multiplicity and completeness of “systems of generalized eigenvectors” have been defined and studied in [51] for a general Banach space operator.

We turn now to the proofs. We return to the functional model described in the previous sections. Henceforth we adopt the notation $W_{0}x=W_{0,\widehat{\mathbb{C}}}\,x,$ where $W_{0,\widehat{\mathbb{C}}}$ is the transform defined in (16) for the case of $\Omega=\widehat{\mathbb{C}}$. We set

[TABLE]

Notice that for any $x$ in $L^{2}(\mu,H_{0})$,

[TABLE]

Moreover, by  (17),

[TABLE]

We will use the space $\mathcal{O}(\mathcal{D},\operatorname{Mod}(\widehat{\mathbb{C}}))=\mathcal{O}(\mathcal{D})\otimes\operatorname{Mod}(\widehat{\mathbb{C}})$. For a function $f(\lambda,z)$ in this space, $f(\lambda,\cdot)$ is an element of $\operatorname{Mod}(\widehat{\mathbb{C}})$ for any $\lambda\in\mathcal{D}$ and the map $\lambda\mapsto f(\lambda,\cdot)\in\operatorname{Mod}(\widehat{\mathbb{C}})$ is analytic. We start by the following observation.

Lemma 7.10**.**

Let $\mathcal{D}$ be any domain in $\mathbb{C}$ and let $\mathcal{D}_{0}$ be its subdomain with admissible boundary such that $\overline{\mathcal{D}}_{0}\subset\mathcal{D}$. Then the restriction-to-diagonal map

[TABLE]

is well defined and continuous. It sends to [math] any function of the form $(z-\lambda)f(\lambda,z)$, where $f(\lambda,z)\in\mathcal{O}(\mathcal{D},\operatorname{Mod}(\widehat{\mathbb{C}}))$.

Proof.

Let $\mathcal{D}_{1}$ be a domain with smooth boundary, relatively compact in $\mathcal{D}$ and containing the closure of $\mathcal{D}_{0}$. Due to the nuclearity of $\mathcal{O}(U)$, for any open set $U$, the restriction map factors through the complete projective tensor product

[TABLE]

Above $L^{2}_{a}(\mathcal{D}_{1})$ stands for the Bergman space.

Choose an orthonormal basis $\{f_{n}(\lambda)\}$ of $L^{2}_{a}(\mathcal{D}_{1})$ such that any $f\in\mathcal{O}(\mathcal{D})$ expands when restricted to $\mathcal{D}_{1}$ into a convergent series $\sum c_{n}f_{n}(\lambda)$. Thus, after taking restrictions to $\mathcal{D}_{1}$ of the elements of $\mathcal{O}(\mathcal{D})\otimes{\rm Mod}(\widehat{\mathbb{C}})$ we can work with convergent series with respect to the projective norm:

[TABLE]

where $g_{n}(z)\in\operatorname{Mod}(\widehat{\mathbb{C}})$. Moreover, the maps $f\in L^{2}_{a}(\mathcal{D}_{1},\operatorname{Mod}(\widehat{\mathbb{C}}))\mapsto g_{n}|\mathcal{D}_{0}$ are bounded and their norms decay exponentially as $n\to\infty$. This implies that the above restriction-to-diagonal operator is bounded. If $f(\lambda,z)$ is expressed in a series as above, then the restriction-to-diagonal operator sends both functions $zf(\lambda,z)$ and $\lambda f(\lambda,z)$ to $\sum zf_{n}(z)g_{n}(z)$, which implies the last statement. ∎

Proof of Theorem 7.5.

We have to prove that for every domain $\mathcal{D}$ in $\mathbb{C}$ and every sequence of $L^{2}(\mu,H_{0})$-valued analytic functions $x_{n}(\lambda)$ defined for $\lambda\in\mathcal{D}$, subject to

[TABLE]

with respect to the topology of $\mathcal{O}(\mathcal{D},L^{2}(\mu,H_{0}))$ satisfies

[TABLE]

Notice that (37) rewrites as

[TABLE]

We already know that normal operators (and more general, all decomposable operators) have property $(\beta)$. Hence it is sufficient to prove that

[TABLE]

Set $f_{n}(\lambda,\cdot)=W_{0}x_{n}(\lambda)$.

By applying the isomorphism $W_{0}$ to (37) and using (35), one finds

[TABLE]

Put $a_{n}(\lambda)=\langle x_{n}(\lambda),v\rangle$. Take any compact subset $K$ of $\mathcal{D}$. There exists a domain $\mathcal{D}_{0}$ with admissible boundary such that $K\subset\mathcal{D}_{0}\subset\overline{\mathcal{D}}_{0}\subset\mathcal{D}$. By virtue of the above Lemma,

[TABLE]

Then

[TABLE]

as $n\to\infty$. This implies $a_{n}\to 0$ in $C(K)\otimes\mathbb{C}^{m}$. We conclude that relation  (39) holds. ∎

Proof of Theorem 7.6.

We prove that the function $\tilde{\psi}$ has no deep zeros. First, observe that for any two $m\times m$ invertible matrices $A$ and $B$, formula $B^{-1}-A^{-1}=A^{-1}(A-B)B^{-1}$ implies

[TABLE]

By passing to a limit, we extend the validity of this inequality to arbitrary $m\times m$ matrices $A$ and $B$. This implies $\tilde{\psi}$ is Hölder-$\alpha$ whenever the matrix-valued function $\Psi$ has the same Hölder exponent.

Assume $|\tilde{\psi}(z)-\tilde{\psi}(w)|\leq C_{0}|z-w|^{\alpha}$ for all $z,w\in\mathbb{C}$. Fix a bounded domain $\mathcal{D}$ and its compact subset $K$ and take an arbitrary function $f$, analytic in $\mathcal{D}$; we have to check (33). Since $|\tilde{\psi}|>\delta>0$ on the complement of $B(0,R)$, we may assume that $\mathcal{D}\subset B(0,R)$. Choose a compact set $K_{0}$ such that $K\subset K_{0}^{o}\subset K_{0}\subset\mathcal{D}$. The distance $d_{0}:=\operatorname{dist}(K_{0},\partial\mathcal{D})$ is positive. Put

[TABLE]

Let $z\in K_{0}$. Then any point $w\in B(z,r_{0}(z))$ belongs to $\mathcal{D}$ and

[TABLE]

Hence $|\tilde{\psi}(w)|\geq|\tilde{\psi}(z)|/2$ for all $w\in B(z,r_{0}(z))$. Consequently

[TABLE]

Hence the subharmonic function $\log|f|$ satisfies $\log|f|\leq C^{\prime}\log(e+|\tilde{\psi}|^{-1})$ on $K_{0}$. By applying Theorem 1 of [11] (with $\alpha=n=2$), we deduce that condition (34) ensures the local boundedness of $f$. ∎

If (D*′*) holds and the measure $\mu$ satisfies an estimate $\mu(Q)\leq C|Q|^{\sigma}$ for all squares $Q$, where $\sigma\in(1,2]$ and $C$ are constants, then $\Psi$ is a Hölder-$\alpha$ function for some positive $\alpha$. This is shown along the lines of the last part of the proof of Lemma 2.3, using the estimate (21) with some minor modifications. We leave the details to the reader.

Remark 7.11**.**

As mentioned above, the case when the “dimension” of $\mu$ is one is the most difficult for our approach. However, if the spectrum of $N$ lies on a smooth curve and $T-N$ is compact, belongs to the Matsaev class and the spectrum of $T$ does not “fill in” the interior of the curve, then, by a result by Radjabalipour and Radjavi [41], $T$ is decomposable. This fails for larger compact perturbations, see [25]. We also refer to [35] for a study of the property $(\beta)$ and decomposability of unilateral and bilateral weighted shifts; the later can be viewed as perturbations of the unweighted bilateral shift, which is unitary. We also can mention that functional models for perturbations of normal operators with spectrum on a straight line or on a curve have been devised by Naboko [36] and by Tikhonov [45].

8. FINAL REMARKS

The present article leaves open a few strings.

Question 8.1**.**

Is it possible to give sufficient conditions for dissectability of $T$ along a curve $\gamma$, which can intersect the spectrum of $T$ in a set of positive length, which would be applicable to an arbitrary measure $\mu$, at least for sufficiently smooth perturbations, in a sense to be made precise?

Our construction does not apply to non-dissectible measures $\mu$. Notice, however, that a simplest and most representative measure $\mu$ of this type is one that is absolutely continuous with respect to $\mathcal{H}^{1}$, restricted to a smooth curve. For this case, any finite rank perturbation of $N$ is decomposable, due to the above-cited paper [41]. This is, in fact, much better behavior than in our case, because no conditions on the finite rank perturbation are necessary. So we ask whether it is possible to merge these two cases and to design a technique which would work for arbitrary measures.

Question 8.2**.**

Let $N$ be a normal operator and let $\mu$ be its scalar spectral measure. For which measures $\mu$, can one prove that any finite rank perturbation of $N$ has a nontrivial invariant subspace?

Question 8.3**.**

If $T$ is a finite rank perturbation of a normal operator, is it true that $T$ has property $(\beta)$ if and only if this is true for $T^{*}$? Is it true for a suitable subclass of finite rank perturbations of normals?

Question 8.4**.**

Give necessary conditions and sufficient conditions for the perturbation $T$ to be similar to a normal operator.

Notice that Theorems 7.5 and 7.6 imply a sort of necessary condition (any operator similar to normal is decomposable). However, as examples show, what we obtain is very far from optimal.

For the case studied in [49], it has been proved there that $T$ is similar to $N$ if and only if $\psi\neq 0$ everywhere on $\mathbb{C}$. However, this cannot be true, for instance, if $\mu$ is a discrete measure.

Question 8.5**.**

Given an operator $N$, the corresponding measure $\mu$ and one more measure $\nu$ on the plane, when is it possible to find a finite rank perturbation $T$ of $N$, similar to a normal operator $N_{1}$, whose scalar spectral measure is $\nu$?

We refer to [37] for answers in the case of rank one perturbations and selfadjoint operators $N$ and $N_{1}$. Notice that compact perturbations of normal operators that are normal themselves is a particular case of Voiculescu’s study [47] of the scattering theory for commuting tuples of selfadjoint operators.