On the regularity of generalized random spectral measures and their applications

Abstract

This paper is devoted to the regularity of generalized random spectral measures. First, we prove that every generalized random spectral measure defined on a locally compact metric space with values in a separable Hilbert space is regular. A solution of the Schrödinger-type random equation is obtained as an application. In the second part of this paper, we show that every finitely additive generalized random spectral measure defined on an arbitrary measurable space with values in a finite-dimensional Hilbert space is also regular. In addition, an random version of Jordan’s classical decomposition theorem for a matrix is provided. Subject Classification. Primary 60H25; Secondary 46G10, 60B11, 45R05.