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Erschienen in:
Preliminaries Kapitel lesen Erstes Kapitel lesen

2018 | OriginalPaper | Buchkapitel

2. Preparatory Concepts

verfasst von : Piotr Budzyński, Zenon Jabłoński, Il Bong Jung, Jan Stochel

Erschienen in: Unbounded Weighted Composition Operators in L²-Spaces

Verlag: Springer International Publishing

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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).

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Vorheriges Kapitel Preliminaries
Nächstes Kapitel Subnormality: General Criteria
Fußnoten
1
Because of Lemma 6 and Proposition 10, the rational function appearing on the right-hand side of the equality in (2.14) takes complex values a.e. [μ]. What is important here is that \(\tilde w\) satisfies the equality \(\{\tilde w= 0\} = \{w=0\}\) a.e. [μ].
 
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Metadaten
Titel
Preparatory Concepts
verfasst von
Piotr Budzyński
Zenon Jabłoński
Il Bong Jung
Jan Stochel
Copyright-Jahr
2018
Buch
Unbounded Weighted Composition Operators in L²-Spaces

Print ISBN: 978-3-319-74038-6

Electronic ISBN: 978-3-319-74039-3

Copyright-Jahr: 2018

DOI
https://doi.org/10.1007/978-3-319-74039-3_2

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