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2018 | OriginalPaper | Chapter

3. Subnormality: General Criteria

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

Published in: Unbounded Weighted Composition Operators in L²-Spaces

Publisher: Springer International Publishing

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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).

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Footnotes
1
By Lemma 6 and (3.2), the right-hand side of the equality in (CC) is an \(\mathscr A\)-measurable function defined a.e. [μ w ].
 
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Metadata
Title
Subnormality: General Criteria
Authors
Piotr Budzyński
Zenon Jabłoński
Il Bong Jung
Jan Stochel
Copyright Year
2018
Book
Unbounded Weighted Composition Operators in L²-Spaces

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

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

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-74039-3_3

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