Bounds for eigenvalues of Schatten-von Neumann operators via self-commutators

Michael Gil'

doi:10.1016/j.jfa.2014.06.019

Bounds for eigenvalues of Schatten-von Neumann operators via self-commutators

Michael Gil'

Department of Mathematics

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

8 Scopus citations

Abstract

Let H be a separable Hilbert space, A be a Schatten-von Neumann operator in H with the finite norm N_2p(A)=[Trace(AA*)p]^1/2p for an integer p≥1 and N1(T)=Trace(TT*)^1/2 be the trace norm of a trace operator T. It is proved that where λ_k(A) (k=1, 2, ...) are the eigenvalues of A, [A,A*]_p=A^p(A*)^p-(A*)^pA^p; A^* is the adjoint to A. This results refines the classical inequality. Lower bounds for N₁([A,A*]_p) are also suggested. In addition, if A is a Hilbert-Schmidt operator, we improve the well-known inequality, where A_I=(A-A^*)/2i.

Original language	English
Pages (from-to)	3500-3506
Number of pages	7
Journal	Journal of Functional Analysis
Volume	267
Issue number	9
DOIs	https://doi.org/10.1016/j.jfa.2014.06.019
State	Published - 1 Nov 2014

Keywords

Inequality for eigenvalues
Schatten-von Neumann operators

ASJC Scopus subject areas

Analysis

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10.1016/j.jfa.2014.06.019

Cite this

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title = "Bounds for eigenvalues of Schatten-von Neumann operators via self-commutators",

abstract = "Let H be a separable Hilbert space, A be a Schatten-von Neumann operator in H with the finite norm N2p(A)=[Trace(AA*)p]1/2p for an integer p≥1 and N1(T)=Trace(TT*)1/2 be the trace norm of a trace operator T. It is proved that where λk(A) (k=1, 2, ...) are the eigenvalues of A, [A,A*]p=Ap(A*)p-(A*)pAp; A* is the adjoint to A. This results refines the classical inequality. Lower bounds for N1([A,A*]p) are also suggested. In addition, if A is a Hilbert-Schmidt operator, we improve the well-known inequality, where AI=(A-A*)/2i.",

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N2 - Let H be a separable Hilbert space, A be a Schatten-von Neumann operator in H with the finite norm N2p(A)=[Trace(AA*)p]1/2p for an integer p≥1 and N1(T)=Trace(TT*)1/2 be the trace norm of a trace operator T. It is proved that where λk(A) (k=1, 2, ...) are the eigenvalues of A, [A,A*]p=Ap(A*)p-(A*)pAp; A* is the adjoint to A. This results refines the classical inequality. Lower bounds for N1([A,A*]p) are also suggested. In addition, if A is a Hilbert-Schmidt operator, we improve the well-known inequality, where AI=(A-A*)/2i.

AB - Let H be a separable Hilbert space, A be a Schatten-von Neumann operator in H with the finite norm N2p(A)=[Trace(AA*)p]1/2p for an integer p≥1 and N1(T)=Trace(TT*)1/2 be the trace norm of a trace operator T. It is proved that where λk(A) (k=1, 2, ...) are the eigenvalues of A, [A,A*]p=Ap(A*)p-(A*)pAp; A* is the adjoint to A. This results refines the classical inequality. Lower bounds for N1([A,A*]p) are also suggested. In addition, if A is a Hilbert-Schmidt operator, we improve the well-known inequality, where AI=(A-A*)/2i.

KW - Inequality for eigenvalues

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Bounds for eigenvalues of Schatten-von Neumann operators via self-commutators

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Keywords

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