Constructing Well-bounded Operators not of type (B) on a Class of Inductive Limits

Well-bounded operators are linear operators on a Banach space \(X\) that have an \(AC[a,b]\) functional calculus for some interval \([a,b]\). A well-bounded operator is of type (B) if it can be written as an integral against a spectral family of projections, and this is always the case when \(X\) is...

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Bibliographic Details
Published in:arXiv.org 2022-08
Main Author: Stoneham, Alan
Format: Article
Language:English
Subjects:
Banach spaces
Linear operators
Operators (mathematics)
Online Access:Get full text
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Summary:Well-bounded operators are linear operators on a Banach space \(X\) that have an \(AC[a,b]\) functional calculus for some interval \([a,b]\). A well-bounded operator is of type (B) if it can be written as an integral against a spectral family of projections, and this is always the case when \(X\) is reflexive. There are many examples of well-bounded operators on non-reflexive spaces that are not of type (B), and it is open whether there is a non-reflexive Banach space upon which every well-bounded operator is of type (B). The spaces constructed by Pisier, which answered a conjecture of Grothendieck in the negative, have been suggested by Cheng and Doust as a candidate to answer this open problem. In this paper, it will be shown that on a class of Banach spaces containing these spaces, there is always a well-bounded operator not of type (B).
ISSN:2331-8422