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arXiv · 1909.06156

The spectrum of a random operator is a random set

Gunnar Taraldsen
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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

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Gunnar Taraldsen. 2019-09-13. The spectrum of a random operator is a random set. https://arxiv.org/abs/1909.06156

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