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Unbounded Weighted Composition Operators in L²-Spaces

Authors: Ph.D. Piotr Budzyński, Ph.D. Zenon Jabłoński, Prof. Il Bong Jung, Prof. Dr. Jan Stochel

Publisher: Springer International Publishing

Book Series : Lecture Notes in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book establishes the foundations of the theory of bounded and unbounded weighted composition operators in L²-spaces. It develops the theory in full generality, meaning that the corresponding composition operators are not assumed to be well defined. A variety of seminormality properties of unbounded weighted composition operators are characterized.

The first-ever criteria for subnormality of unbounded weighted composition operators are provided and the subtle interplay between the classical moment problem, graph theory and the injectivity problem for weighted composition operators is revealed. The relationships between weighted composition operators and the corresponding multiplication and composition operators are investigated. The optimality of the obtained results is illustrated by a variety of examples, including those of discrete and continuous types.

The book is primarily aimed at researchers in single or multivariable operator theory.

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Table of Contents

Frontmatter

Chapter 1. Preliminaries

Abstract

The classical Banach-Stone theorem (see [7, Théorème XI.3] and [146], see also [68, Theorem 2.1.1]) states that if X and Y are compact Hausdorff topological spaces and A: C(X) → C(Y ) is a surjective linear isometry, then there exist a continuous function \(w\colon Y\to \mathbb R\) and a homeomorphism ϕ: Y → X such that |w|≡ 1 and

Piotr Budzyński, Zenon Jabłoński, Il Bong Jung, Jan Stochel

Chapter 2. Preparatory Concepts

Abstract

This chapter introduces some concepts of measure theory that will be useful for studying weighted composition operators (including the Radon-Nikodym derivative h _ϕ,w and the conditional expectation \(\mathsf {E}(\cdot \,;\phi ^{-1}(\mathscr A),\mu _w)\); see Sects. 2.1 and 2.4). Weighted composition operators are introduced and initially investigated in Sect. 2.2. Assorted classes of weighted composition operators including classical (unilateral and bilateral) weighted shifts and their adjoints are discussed in Sect. 2.3. The polar decompositions of a weighted composition operator and its adjoint are explicitly described in Sect. 2.5. In Sect. 2.6, characterizations of the quasinormality of weighted composition operators are given (see Theorem 20).

Piotr Budzyński, Zenon Jabłoński, Il Bong Jung, Jan Stochel

Chapter 3. Subnormality: General Criteria

Abstract

The main goal of this chapter is to provide criteria for the subnormality of (not necessarily bounded) weighted composition operators. The first criterion, which is given in Sect. 3.1, requires that h _ϕ,w > 0 a.e. [μ _w] and that there exists a measurable family of Borel probability measures on \(\mathbb R_+\) satisfying the consistency condition (CC) (see Theorem 29). Section 3.3 provides the second criterion which involves another, stronger than (CC), condition (CC⁻¹) (see Theorem 34). In Sect. 3.4, we discuss the interplay between the conditions (CC) and (CC⁻¹) (see Theorem 40). Section 3.2 shows that the consistency condition (CC) itself is not sufficient for subnormality even in the case of composition operators. By Theorem 34, this means that (CC) does not imply (CC⁻¹).

Piotr Budzyński, Zenon Jabłoński, Il Bong Jung, Jan Stochel

Chapter 4. C ∞ -Vectors

Abstract

In this chapter, we turn our interest to weighted composition operators that have sufficiently many

Piotr Budzyński, Zenon Jabłoński, Il Bong Jung, Jan Stochel

Chapter 5. Seminormality

Abstract

In this chapter, we give characterizations of seminormal, formally normal, symmetric, selfadjoint and positive selfadjoint weighted composition operators. Hyponormality and cohyponormality are characterized in Sects. 5.1 and 5.2, respectively (see Theorems 53 and 60). The introductory part of Sect. 5.2 is devoted to the study of the range of the conditional expectation E _ϕ,w. In Sect. 5.3, we characterize normal weighted composition operators (see Theorem 63). We also show that formally normal (in particular, symmetric) weighted composition operators are automatically normal (see Theorem 66). In Sect. 5.4, we characterize selfadjoint and positive selfadjoint weighted composition operators (see Theorems 72 and 76).

Piotr Budzyński, Zenon Jabłoński, Il Bong Jung, Jan Stochel

Chapter 6. Discrete Measure Spaces

Abstract

In this chapter, we adapt our general results to the context of discrete weighted composition operators, i.e., weighted composition operators over discrete measure spaces. Section 6.1 has an introductory character. Section 6.2 characterizes the hyponormality, cohyponormality and normality of discrete weighted composition operators (see Theorems 83, 84 and 87). Section 6.3 provides two criteria for the subnormality of discrete weighted composition operators, the second of which generalizes the discrete version of one of Lambert’s characterizations of bounded subnormal composition operators (see Theorems 89 and 90). The interplay between the theory of moments, the geometry of graphs induced by symbols and the injectivity problem is discussed in Section 6.4 (see Theorem 93 and Problems 96 and 100). Section 6.5 contains a variety of examples illustrating our considerations.

Piotr Budzyński, Zenon Jabłoński, Il Bong Jung, Jan Stochel

Chapter 7. Relationships Between Cϕ,w and C ϕ

Abstract

In this chapter, we investigate the interplay between selected properties of a weighted composition operator C _ϕ,w and the corresponding composition operator C _ϕ. In Sect. 7.1, we discuss the questions of when the product M _wC _ϕ is closed and when it coincides with C _ϕ,w (see Theorems 110 and 112). The relationships between the Radon-Nikodym derivatives h _ϕ and h _ϕ,w are described in Sect. 7.2 (see Propositions 116, 119, and 121). In Sect. 7.3, using a result due to Berg and Durán, we give conditions enabling us to deduce the subnormality of C _ϕ,w from that of C _ϕ (see Theorem 126). The converse possibility is discussed in Theorem 130. In Sect. 7.4, we provide a criterion for a bounded weighted composition operator with matrix symbol to be subnormal (see Theorem 131). Section 7.5 contains numerous examples illustrating our considerations.

Piotr Budzyński, Zenon Jabłoński, Il Bong Jung, Jan Stochel

Chapter 8. Miscellanea

Abstract

This chapter consists of three sections. In Section 8.1, we discuss the problem of whether the tensor product of (finitely many) weighted composition operators can be regarded as a weighted composition operator. We begin by investigating the question of when the well-definiteness of \(C_{\phi _i,w_i}\), i = 1, …, N, implies the well-definiteness of C _ϕ,w, where ϕ = ϕ ₁ ×… × ϕ _N and w = w ₁ ⊗… ⊗ w _N (see Theorem 149 and Corollary 151). In Theorem 154 we show that the closure of the tensor product \(C_{\phi _1,w_1} \otimes \ldots \otimes C_{\phi _N,w_N}\) of densely defined weighted composition operators can be regarded as the weighted composition operator C _ϕ,w. Two open questions related to the above topics are stated as well (see Problems 146 and 155). Section 8.2 proposes a method of modifying the symbol ϕ of a weighted composition operator C _ϕ,w which preserves many properties of objects attached to C _ϕ,w and does not change the operator C _ϕ,w itself. As shown in Section 8.3, this method enables us to modify the symbol ϕ of a quasinormal weighted composition operator C _ϕ,w so as to get a \(\phi ^{-1}(\mathcal A)\)-measurable family \(P\colon X \times {\mathfrak B}(\mathcal R_+) \to [0,1]\) of probability measures that satisfies (CC−1) (see Proposition 161). We conclude Section 8.3 with an example of a quasinormal weighted composition operator C _ϕ,w which has no \(\phi ^{-1}(\mathcal A)\)-measurable family P of probability measures on \(\mathcal R_+\) satisfying (CC) (see Example 162).

Piotr Budzyński, Zenon Jabłoński, Il Bong Jung, Jan Stochel

Backmatter

Title: Unbounded Weighted Composition Operators in L²-Spaces
Authors: Ph.D. Piotr Budzyński
Ph.D. Zenon Jabłoński
Prof. Il Bong Jung
Prof. Dr. Jan Stochel
Copyright Year: 2018
Publisher: Springer International Publishing
Electronic ISBN: 978-3-319-74039-3
Print ISBN: 978-3-319-74038-6
DOI: https://doi.org/10.1007/978-3-319-74039-3

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