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A Measurable Selector in Kadison’s Carpenter’s Theorem

Part of: Normed linear spaces and Banach spaces; Banach lattices Nontrigonometric harmonic analysis Special classes of linear operators

Published online by Cambridge University Press:  16 July 2019

Marcin Bownik  and
Marcin Szyszkowski
Marcin Bownik
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97403–1222, USA Email: mbownik@uoregon.edu
Marcin Szyszkowski
Affiliation:
Faculty of Applied Physics and Mathematics, Department of Probability and Biomathematics,Gdańsk Univerity of Technology, ul. Narutowicza 11/12, 80-233Gdańsk, Poland Email: fox@mat.ug.edu.pl
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Abstract

We show the existence of a measurable selector in Carpenter’s Theorem due to Kadison. This solves a problem posed by Jasper and the first author in an earlier work. As an application we obtain a characterization of all possible spectral functions of shift-invariant subspaces of $L^{2}(\mathbb{R}^{d})$ and Carpenter’s Theorem for type $\text{I}_{\infty }$ von Neumann algebras.

Keywords

Schur–Horn problemdiagonal of self-adjoint operatorCarpenter’s theorem

MSC classification

Primary: 42C15: General harmonic expansions, frames
Secondary: 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.) 46C05: Hilbert and pre-Hilbert spaces
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 6 , December 2020 , pp. 1505 - 1528
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

Author M. B. was supported in part by the NSF grant DMS-1665056.

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