Perturbation of operators and approximation of spectrum

TL;DR

This paper develops a method to approximate the spectrum of holomorphic families of bounded self-adjoint operators using finite-dimensional compressions, extending known results and exploring spectral gap predictions.

Contribution

It extends spectral approximation techniques to holomorphic operator families and investigates spectral gap prediction through a pure linear algebra approach.

Findings

Spectral bounds can be uniformly approximated on compact sets.

Eigenvalue functions of finite-dimensional compressions converge to spectral bounds.

The approach applies to block Toeplitz-Laurent operators.

Abstract

Let A(x) be a holomorphic family of bounded self-adjoint operators on a separable Hilbert space H and let A(x)_n be the orthogonal compressions of A(x) to the span of first n elements of an orthonormal basis of H. The problem considered here is to approximate the spectrum of A(x) using the sequence of eigenvalues of A(x)_n. We show that the bounds of the essential spectrum and the discrete spectral values outside the bounds of essential spectrum of A(x) can be approximated uniformly on all compact subsets by the sequence of eigenvalue functions of A(x)_n. The known results for a bounded selfadjoint operator, are translated into the case of a holomorphic family of operators. Also an attempt is made to predict the existence of spectral gaps that may occur between the bounds of essential spectrum of A(0) = A and study the effect of holomorphic perturbation of operators in the prediction of spectral gaps. As an example, gap issues of some block Toeplitz-Laurent operators are discussed. The pure linear algebraic approach is the main advantage of the results here.