On matching distance between eigenvalues of unbounded operators

Michael Gil

doi:10.33205/cma.1060718

On matching distance between eigenvalues of unbounded operators

Michael Gil

Department of Mathematics

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

4 Scopus citations

Abstract

Let A and Ã be linear operators on a Banach space having compact resolvents, and let λ_k(A) and λ_k(Ã) (k = 1, 2, ...) be the eigenvalues taken with their algebraic multiplicities of A and Ã, respectively. Under some conditions, we derive a bound for the quantity md(A, Ã):= inf sup |λ_π(k₎(Ã) − λ_k(A)|, ^π k=1,2,... where π is taken over all permutations of the set of all positive integers. That quantity is called the matching optimal distance between the eigenvalues of A and Ã. Applications of the obtained bound to matrix differential operators are also discussed.

Original language	English
Pages (from-to)	46-53
Number of pages	8
Journal	Constructive Mathematical Analysis
Volume	5
Issue number	1
DOIs	https://doi.org/10.33205/cma.1060718
State	Published - 1 Jan 2022

Keywords

Banach space
differential operator
matching distance
perturbations of eigenvalues
tensor product of Hilbert spaces

ASJC Scopus subject areas

Analysis
Numerical Analysis
Applied Mathematics

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10.33205/cma.1060718

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N2 - Let A and Ã be linear operators on a Banach space having compact resolvents, and let λk(A) and λk(Ã) (k = 1, 2, ...) be the eigenvalues taken with their algebraic multiplicities of A and Ã, respectively. Under some conditions, we derive a bound for the quantity md(A, Ã):= inf sup |λπ(k)(Ã) − λk(A)|, π k=1,2,... where π is taken over all permutations of the set of all positive integers. That quantity is called the matching optimal distance between the eigenvalues of A and Ã. Applications of the obtained bound to matrix differential operators are also discussed.

AB - Let A and Ã be linear operators on a Banach space having compact resolvents, and let λk(A) and λk(Ã) (k = 1, 2, ...) be the eigenvalues taken with their algebraic multiplicities of A and Ã, respectively. Under some conditions, we derive a bound for the quantity md(A, Ã):= inf sup |λπ(k)(Ã) − λk(A)|, π k=1,2,... where π is taken over all permutations of the set of all positive integers. That quantity is called the matching optimal distance between the eigenvalues of A and Ã. Applications of the obtained bound to matrix differential operators are also discussed.

KW - Banach space

KW - differential operator

KW - matching distance

KW - perturbations of eigenvalues

KW - tensor product of Hilbert spaces

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DO - 10.33205/cma.1060718

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SP - 46

EP - 53

JO - Constructive Mathematical Analysis

JF - Constructive Mathematical Analysis

IS - 1

ER -

On matching distance between eigenvalues of unbounded operators

Abstract

Keywords

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