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: Contribution to journal › Article › peer-review

Abstract

Given several bounded linear operators A₁,.. , A_n on a Hilbert space, their projective spectrum is the set of complex vectors z= (z₁,.. , z_n) such that the multiparameter pencil A(z) = z₁A₁+ ⋯ + z_nA_n 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 La2(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
Article number	81
Journal	Complex Analysis and Operator Theory
Volume	17
Issue number	6
DOIs	https://doi.org/10.1007/s11785-023-01383-3
State	Published - Sep 2023

Keywords

Projective spectrum
Shift operator
Toeplitz operator

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title = "A Note on Joint Spectrum in Function Spaces",

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+ ⋯ + znAn 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 La2(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.",

keywords = "Projective spectrum, Shift operator, Toeplitz operator",

author = "Puyu Cui and Rongwei Yang",

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year = "2023",

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doi = "10.1007/s11785-023-01383-3",

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AU - Cui, Puyu

AU - Yang, Rongwei

PY - 2023/9

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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+ ⋯ + znAn 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 La2(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+ ⋯ + znAn 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 La2(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.

KW - Projective spectrum

KW - Shift operator

KW - Toeplitz operator

UR - https://www.scopus.com/pages/publications/85165307316

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DO - 10.1007/s11785-023-01383-3

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VL - 17

JO - Complex Analysis and Operator Theory

JF - Complex Analysis and Operator Theory

IS - 6

M1 - 81

ER -

A Note on Joint Spectrum in Function Spaces

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