Compact operators whose real and imaginary parts are positive

Rajendra Bhatia; Xingzhi Zhan

doi:10.1090/s0002-9939-00-05832-9

Compact operators whose real and imaginary parts are positive

Rajendra Bhatia, Xingzhi Zhan

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

18 Scopus citations

Abstract

Let T be a compact operator on a Hilbert space such that the operators A = 1/2(T + T*) and B = 1/2i(T - T*) are positive. Let {s_j} be the singular values of T and {α_j}, {β_j} the eigenvalues of A, B, all enumerated in decreasing order. We show that the sequence {s_j²} is majorised by {α_j² + β_j²}. An important consequence is that, when p ≥ 2, ∥T∥_p² is less than or equal to ∥A∥_p² + ∥B∥_p², and when 1 ≤ p ≤ 2, this inequality is reversed.

Original language	English
Pages (from-to)	2277-2281
Number of pages	5
Journal	Proceedings of the American Mathematical Society
Volume	129
Issue number	8
DOIs	https://doi.org/10.1090/s0002-9939-00-05832-9
State	Published - 2001
Externally published	Yes

Keywords

Compact operator
Eigenvalues
Majorisation
Positive operator
Schatten p-norms
Singular values

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10.1090/s0002-9939-00-05832-9

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title = "Compact operators whose real and imaginary parts are positive",

abstract = "Let T be a compact operator on a Hilbert space such that the operators A = 1/2(T + T*) and B = 1/2i(T - T*) are positive. Let \{sj\} be the singular values of T and \{αj\}, \{βj\} the eigenvalues of A, B, all enumerated in decreasing order. We show that the sequence \{sj2\} is majorised by \{αj2 + βj2\}. An important consequence is that, when p ≥ 2, ∥T∥p2 is less than or equal to ∥A∥p2 + ∥B∥p2, and when 1 ≤ p ≤ 2, this inequality is reversed.",

keywords = "Compact operator, Eigenvalues, Majorisation, Positive operator, Schatten p-norms, Singular values",

author = "Rajendra Bhatia and Xingzhi Zhan",

year = "2001",

doi = "10.1090/s0002-9939-00-05832-9",

language = "英语",

volume = "129",

pages = "2277--2281",

journal = "Proceedings of the American Mathematical Society",

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T1 - Compact operators whose real and imaginary parts are positive

AU - Bhatia, Rajendra

AU - Zhan, Xingzhi

PY - 2001

Y1 - 2001

N2 - Let T be a compact operator on a Hilbert space such that the operators A = 1/2(T + T*) and B = 1/2i(T - T*) are positive. Let {sj} be the singular values of T and {αj}, {βj} the eigenvalues of A, B, all enumerated in decreasing order. We show that the sequence {sj2} is majorised by {αj2 + βj2}. An important consequence is that, when p ≥ 2, ∥T∥p2 is less than or equal to ∥A∥p2 + ∥B∥p2, and when 1 ≤ p ≤ 2, this inequality is reversed.

AB - Let T be a compact operator on a Hilbert space such that the operators A = 1/2(T + T*) and B = 1/2i(T - T*) are positive. Let {sj} be the singular values of T and {αj}, {βj} the eigenvalues of A, B, all enumerated in decreasing order. We show that the sequence {sj2} is majorised by {αj2 + βj2}. An important consequence is that, when p ≥ 2, ∥T∥p2 is less than or equal to ∥A∥p2 + ∥B∥p2, and when 1 ≤ p ≤ 2, this inequality is reversed.

KW - Compact operator

KW - Eigenvalues

KW - Majorisation

KW - Positive operator

KW - Schatten p-norms

KW - Singular values

UR - https://www.scopus.com/pages/publications/23044526198

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DO - 10.1090/s0002-9939-00-05832-9

M3 - 文章

AN - SCOPUS:23044526198

SN - 0002-9939

VL - 129

SP - 2277

EP - 2281

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 8

ER -

Compact operators whose real and imaginary parts are positive

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