Ergodic Families of Self-Adjoint Operators

Abstract

Throughout this chapter (Ω,F) is a fixed measurable space and H is a separable complex Hilbert space on which all the operators to be considered are defined. We want to investigate the measurability properties of operator valued functions of the form:

$$ \omega \; \to H(\omega ) $$

from Ω into the space of operators on H We are mainly interested in self-adjoint operators and, as we explained in Chapter I, mostly in are unbounded ones. With this in mind we choose a notion of measurability of operator valued functions which relies on the functional calculus of self-adjoint operators and on the properties of bounded functions of these possibly unbounded operators. We believe that all the technical difficulties relative to these measurability problems have been carefully swept under the rug in most of the research litterature. This is one of the reasons why we decided to study these problems thoroughly.