Perturbations of real parts of eigenvalues of bounded linear operators in a Hilbert space

Michael Gil'

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 2, page 567-573
  • ISSN: 0011-4642

Abstract

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Let A be a bounded linear operator in a complex separable Hilbert space ℋ , and S be a selfadjoint operator in ℋ . Assuming that A - S belongs to the Schatten-von Neumann ideal 𝒮 p ( p > 1 ) , we derive a bound for ∑ k | R λ k ( A ) - λ k ( S ) | p , where λ k ( A ) ( k = 1 , 2 , ⋯ ) are the eigenvalues of A . Our results are formulated in terms of the “extended” eigenvalue sets in the sense introduced by T. Kato. In addition, in the case p = 2 we refine the Weyl inequality between the real parts of the eigenvalues of A and the eigenvalues of its Hermitian component.

How to cite

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Gil', Michael. "Perturbations of real parts of eigenvalues of bounded linear operators in a Hilbert space." Czechoslovak Mathematical Journal 74.2 (2024): 567-573. <http://eudml.org/doc/299544>.

@article{Gil2024,
abstract = {Let $A$ be a bounded linear operator in a complex separable Hilbert space $\mathcal \{H\}$, and $S$ be a selfadjoint operator in $\mathcal \{H\}$. Assuming that $A-S$ belongs to the Schatten-von Neumann ideal $\mathcal \{S\}_p$$(p> 1),$ we derive a bound for $\sum _\{k\}| \{\rm R\} \lambda _k(A)-\lambda _k(S)|^p$, where $\lambda _k(A)$$(k=1, 2, \dots )$ are the eigenvalues of $A$. Our results are formulated in terms of the “extended” eigenvalue sets in the sense introduced by T. Kato. In addition, in the case $p=2$ we refine the Weyl inequality between the real parts of the eigenvalues of $A$ and the eigenvalues of its Hermitian component.},
author = {Gil', Michael},
journal = {Czechoslovak Mathematical Journal},
keywords = {Hilbert space; linear operator; eigenvalue; Kato theorem; Weyl inequality},
language = {eng},
number = {2},
pages = {567-573},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Perturbations of real parts of eigenvalues of bounded linear operators in a Hilbert space},
url = {http://eudml.org/doc/299544},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Gil', Michael
TI - Perturbations of real parts of eigenvalues of bounded linear operators in a Hilbert space
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 2
SP - 567
EP - 573
AB - Let $A$ be a bounded linear operator in a complex separable Hilbert space $\mathcal {H}$, and $S$ be a selfadjoint operator in $\mathcal {H}$. Assuming that $A-S$ belongs to the Schatten-von Neumann ideal $\mathcal {S}_p$$(p> 1),$ we derive a bound for $\sum _{k}| {\rm R} \lambda _k(A)-\lambda _k(S)|^p$, where $\lambda _k(A)$$(k=1, 2, \dots )$ are the eigenvalues of $A$. Our results are formulated in terms of the “extended” eigenvalue sets in the sense introduced by T. Kato. In addition, in the case $p=2$ we refine the Weyl inequality between the real parts of the eigenvalues of $A$ and the eigenvalues of its Hermitian component.
LA - eng
KW - Hilbert space; linear operator; eigenvalue; Kato theorem; Weyl inequality
UR - http://eudml.org/doc/299544
ER -

References

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