The 2×2 block matrices associated with an annulus

Sourav Pal; Nitin Tomar

doi:10.1007/s00013-024-02058-x

The 2×2 block matrices associated with an annulus

Sourav Pal, Nitin Tomar

Research output: Contribution to journal › Article › peer-review

Abstract

A bounded Hilbert space operator T for which the closure of the annulus (Formula presented.) is a spectral set is called an A_r-contraction. A celebrated theorem due to Douglas, Muhly, and Pearcy gives a necessary and sufficient condition such that a 2×2 block matrix of operators T₁X0T₂ is a contraction. We seek an answer to the same question in the setting of an annulus, i.e., under what conditions does T~_Y=T₁Y0T₂ become an A_r-contraction? For A_r-contractions T,T₁,T₂ and an operator X that commutes with T,T₁,T₂, here we find a necessary and sufficient condition such that each of the block matrices (Formula presented.) becomes an A_r-contraction.

Original language	English
Pages (from-to)	75-82
Number of pages	8
Journal	Archiv der Mathematik
Volume	124
Issue number	1
DOIs	https://doi.org/10.1007/s00013-024-02058-x
State	Published - 1 Jan 2025
Externally published	Yes

Keywords

A-contraction
Annulus
Block matrix

ASJC Scopus subject areas

General Mathematics

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10.1007/s00013-024-02058-x

Cite this

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title = "The 2×2 block matrices associated with an annulus",

abstract = "A bounded Hilbert space operator T for which the closure of the annulus (Formula presented.) is a spectral set is called an Ar-contraction. A celebrated theorem due to Douglas, Muhly, and Pearcy gives a necessary and sufficient condition such that a 2×2 block matrix of operators T1X0T2 is a contraction. We seek an answer to the same question in the setting of an annulus, i.e., under what conditions does T\textasciitilde{}Y=T1Y0T2 become an Ar-contraction? For Ar-contractions T,T1,T2 and an operator X that commutes with T,T1,T2, here we find a necessary and sufficient condition such that each of the block matrices (Formula presented.) becomes an Ar-contraction.",

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AU - Pal, Sourav

AU - Tomar, Nitin

N1 - Publisher Copyright: © Springer Nature Switzerland AG 2024.

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N2 - A bounded Hilbert space operator T for which the closure of the annulus (Formula presented.) is a spectral set is called an Ar-contraction. A celebrated theorem due to Douglas, Muhly, and Pearcy gives a necessary and sufficient condition such that a 2×2 block matrix of operators T1X0T2 is a contraction. We seek an answer to the same question in the setting of an annulus, i.e., under what conditions does T~Y=T1Y0T2 become an Ar-contraction? For Ar-contractions T,T1,T2 and an operator X that commutes with T,T1,T2, here we find a necessary and sufficient condition such that each of the block matrices (Formula presented.) becomes an Ar-contraction.

AB - A bounded Hilbert space operator T for which the closure of the annulus (Formula presented.) is a spectral set is called an Ar-contraction. A celebrated theorem due to Douglas, Muhly, and Pearcy gives a necessary and sufficient condition such that a 2×2 block matrix of operators T1X0T2 is a contraction. We seek an answer to the same question in the setting of an annulus, i.e., under what conditions does T~Y=T1Y0T2 become an Ar-contraction? For Ar-contractions T,T1,T2 and an operator X that commutes with T,T1,T2, here we find a necessary and sufficient condition such that each of the block matrices (Formula presented.) becomes an Ar-contraction.

KW - A-contraction

KW - Annulus

KW - Block matrix

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JO - Archiv der Mathematik

JF - Archiv der Mathematik

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ER -

The 2×2 block matrices associated with an annulus

Abstract

Keywords

ASJC Scopus subject areas

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