The spectrum of a random operator is a random set

TL;DR

This paper demonstrates that the spectrum and eigenvalues of random operators, defined via random sets, are themselves random sets, using tools from the theory of random sets and spectral analysis.

Contribution

It introduces a novel approach by defining random operators through their graphs as random sets and proves their spectra are also random sets, even for bounded operators.

Findings

Spectrum of random operators is a random set.

Eigenvalues form a random set.

Applicable to bounded and unbounded operators.

Abstract

The theory of random sets is demonstrated to prove useful for the theory of random operators. A random operator is here defined by requiring the graph to be a random set. It is proved that the spectrum and the set of eigenvalues of random operators are random sets. These results seem to be a novelty even in the case of random bounded operators. The main technical tools are given by the measurable selection theorem, the measurable projection theorem, and a characterisation of the spectrum by approximate eigenvalues of the operator and the adjoint operator. A discussion of some of the existing definitions of the concept of a random operator is included at the end of the paper. Keywords: Random operators; Set-valued functions; General topics in linear spectral theory; Random operators and equations; Stochastic integrals; Disordered systems