Lower bounds for eigenvalues of Schatten-von Neumann operators

M. I. Gil

Lower bounds for eigenvalues of Schatten-von Neumann operators

M. I. Gil

Department of Mathematics

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

11 Scopus citations

Abstract

Let S_p be the Schatten-von Neumann ideal of compact operators equipped with the norm N_p(·). For an A ∈ S_p, (1 < p < ∞), the inequality [∑_k=1^∞ | Re λ_k(A) |^p]^1/p + b_p [∑_k=1^∞ | Im λ_k(A)| ^p]^1/p ≥ N_p(A_R) - b _pN_p(A_I) (b_p = const. > 0) is derived, where λ_j(A) (j = 1, 2, . . .) are the eigenvalues of A, A_I = (A - A*)/2i and A_R = (A + A*)/2. The suggested approach is based on some relations between the real and imaginary Hermitian components of quasinilpotent operators.

Original language	English
Article number	66
Journal	Journal of Inequalities in Pure and Applied Mathematics
Volume	8
Issue number	3
State	Published - 22 Oct 2007

Keywords

Inequalities for eigenvalues
Schatten-von Neumann ideals

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

Cite this

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title = "Lower bounds for eigenvalues of Schatten-von Neumann operators",

abstract = "Let Sp be the Schatten-von Neumann ideal of compact operators equipped with the norm Np(·). For an A ∈ Sp, (1 < p < ∞), the inequality [∑k=1∞ | Re λk(A) |p]1/p + bp [∑k=1∞ | Im λk(A)| p]1/p ≥ Np(AR) - b pNp(AI) (bp = const. > 0) is derived, where λj(A) (j = 1, 2, . . .) are the eigenvalues of A, AI = (A - A*)/2i and AR = (A + A*)/2. The suggested approach is based on some relations between the real and imaginary Hermitian components of quasinilpotent operators.",

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N2 - Let Sp be the Schatten-von Neumann ideal of compact operators equipped with the norm Np(·). For an A ∈ Sp, (1 < p < ∞), the inequality [∑k=1∞ | Re λk(A) |p]1/p + bp [∑k=1∞ | Im λk(A)| p]1/p ≥ Np(AR) - b pNp(AI) (bp = const. > 0) is derived, where λj(A) (j = 1, 2, . . .) are the eigenvalues of A, AI = (A - A*)/2i and AR = (A + A*)/2. The suggested approach is based on some relations between the real and imaginary Hermitian components of quasinilpotent operators.

AB - Let Sp be the Schatten-von Neumann ideal of compact operators equipped with the norm Np(·). For an A ∈ Sp, (1 < p < ∞), the inequality [∑k=1∞ | Re λk(A) |p]1/p + bp [∑k=1∞ | Im λk(A)| p]1/p ≥ Np(AR) - b pNp(AI) (bp = const. > 0) is derived, where λj(A) (j = 1, 2, . . .) are the eigenvalues of A, AI = (A - A*)/2i and AR = (A + A*)/2. The suggested approach is based on some relations between the real and imaginary Hermitian components of quasinilpotent operators.

KW - Inequalities for eigenvalues

KW - Schatten-von Neumann ideals

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M3 - Article

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SN - 1443-5756

VL - 8

JO - Journal of Inequalities in Pure and Applied Mathematics

JF - Journal of Inequalities in Pure and Applied Mathematics

IS - 3

M1 - 66

ER -

Lower bounds for eigenvalues of Schatten-von Neumann operators

Abstract

Keywords

ASJC Scopus subject areas

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