Abstract
The minimax principle for eigenvalues in gaps of the essential spectrum in the form presented by Griesemer et al. (Doc Math 4:275–283, 1999) is adapted to cover certain abstract perturbative settings with bounded or unbounded perturbations, in particular ones that are off-diagonal with respect to the spectral gap under consideration. This in part builds upon and extends the considerations in the author’s appendix to Nakić et al. (J Spectr Theory 10:843–885, 2020). Several monotonicity and continuity properties of eigenvalues in gaps of the essential spectrum are deduced, and the Stokes operator is revisited as an example.
Highlights
Introduction and Main ResultThe standard Courant minimax values λk(A) of a lower semibounded operator A on a Hilbert space H are given by λk( A) = inf sup x, Ax = inf sup a[x, x] M⊂Dom(A) x∈MM⊂Dom(| A|1/2) x∈M dim M=k x =1 dim M=k x =1Communicated by Jussi Behrndt.This article is part of the topical collection “Spectral Theory and Operators in Mathematical Physics” edited by Jussi Behrndt, Fabrizio Colombo and Sergey Naboko. 29 Page 2 of 36A
The above minimax values have proved to be a powerful description of the eigenvalues below the essential spectrum of A; they agree with these eigenvalues in nondecreasing order counting multiplicities as long as the latter exist and else equal the bottom of the essential spectrum
A standard application in this context is that the eigenvalues below the essential spectrum exhibit a monotonicity with respect to the operator: for two lower semibounded self-adjoint operators A and B with A ≤ B in the sense of quadratic forms one has λk( A) ≤ λk(B) for all k, see, e.g., [31, Corollary 12.3]
Summary
This article is part of the topical collection “Spectral Theory and Operators in Mathematical Physics” edited by Jussi Behrndt, Fabrizio Colombo and Sergey Naboko
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