Spectral approximations

Abstract

The problem of determining the nonzero eigenvalues, if any exist, of a linear operator on a Hilbert space, is a very important one in mathematics and physics. Even when the space is finite-dimensional, the actual determination of the eigenvalues can be quite non-trivial. On infinite-dimensional spaces, the problem is even more difficult. There are various special classes of operators, compact or' otherwise, e.g. integral operators with Cauchy-type kernel, Toeplitz, composition operators, and differential operators of diverse types, on such spaces, the spectral problems associated with which are very important from the point of view of applications. It is therefore of interest to attempt to approximate the eigenvalues of a given operator by the eigenvalues, hopefully easier to compute, of a sequence of operators which converge to the given operator in some way that is both natural in the situation at hand and tractable.