A Note on Joint Spectrum in Function Spaces

Puyu Cui; Rongwei Yang

doi:10.1007/s11785-023-01383-3

A Note on Joint Spectrum in Function Spaces

Puyu Cui
, Rongwei Yang

Mathematics and Statistics

University at Albany

Liaoning Normal University

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

Abstract

Given several bounded linear operators A1, An on a Hilbert space, their projective spectrum is the set of complex vectors z = (z1, …, zn) such that the multiparameter pencil A(z) = z1A1 +·, ·+ zn An is not invertible. This paper studies the projective spectrum of the shift operator T, its adjoint T ∗ and a projection operator P. Two spaces of concern are the classical Bergman space L2 a(D) and the L2 space over the torus T2. The projective spectra are completely determined in both cases. The results lead to new questions about Toeplitz operators.

Original language	English
Title of host publication	Multivariable Operator Theory
Subtitle of host publication	the Jörg Eschmeier Memorial Volume
Publisher	Springer Nature
Pages	245-359
Number of pages	115
ISBN (Electronic)	9783031505355
ISBN (Print)	9783031505348
DOIs	https://doi.org/10.1007/s11785-023-01383-3
State	Published - Jan 1 2023

Keywords

Projective spectrum
Shift operator
Toeplitz operator

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10.1007/s11785-023-01383-3

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abstract = "Given several bounded linear operators A1, An on a Hilbert space, their projective spectrum is the set of complex vectors z = (z1, …, zn) such that the multiparameter pencil A(z) = z1A1 +·, ·+ zn An is not invertible. This paper studies the projective spectrum of the shift operator T, its adjoint T ∗ and a projection operator P. Two spaces of concern are the classical Bergman space L2 a(D) and the L2 space over the torus T2. The projective spectra are completely determined in both cases. The results lead to new questions about Toeplitz operators.",

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N2 - Given several bounded linear operators A1, An on a Hilbert space, their projective spectrum is the set of complex vectors z = (z1, …, zn) such that the multiparameter pencil A(z) = z1A1 +·, ·+ zn An is not invertible. This paper studies the projective spectrum of the shift operator T, its adjoint T ∗ and a projection operator P. Two spaces of concern are the classical Bergman space L2 a(D) and the L2 space over the torus T2. The projective spectra are completely determined in both cases. The results lead to new questions about Toeplitz operators.

AB - Given several bounded linear operators A1, An on a Hilbert space, their projective spectrum is the set of complex vectors z = (z1, …, zn) such that the multiparameter pencil A(z) = z1A1 +·, ·+ zn An is not invertible. This paper studies the projective spectrum of the shift operator T, its adjoint T ∗ and a projection operator P. Two spaces of concern are the classical Bergman space L2 a(D) and the L2 space over the torus T2. The projective spectra are completely determined in both cases. The results lead to new questions about Toeplitz operators.

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KW - Shift operator

KW - Toeplitz operator

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BT - Multivariable Operator Theory

PB - Springer Nature

ER -

A Note on Joint Spectrum in Function Spaces

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Keywords

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