2019/09/13 by Taraldsen, Gunnar
#26E25 #35P05 #47B80 #60H05 #60H25 #82B44 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1909.06156
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