Truncation Method For Random Bounded Self-Adjoint Operators

Abstract

This article addresses the following question; 'how to approximate the spectrum of random bounded self-adjoint operators on separable Hilbert spaces'. This is an attempt to establish a link between the spectral theory of random operators and the rich theory of random matrices; including various notions of convergence. This study tries to develop a random version of the truncation method, which is useful in approximating spectrum of bounded self-adjoint operators. It is proved that the eigenvalue sequences of the truncations converge in distribution to the eigenvalues of the random bounded self-adjoint operator. The convergence of moments are also proved with some examples. In addition, the article discusses some new methods to predict the existence of spectral gaps between the bounds of essential spectrum. Some important open problems are also stated at the end.