On a Functional Calculus for Decomposable Operators and Applications to Normal, Operator-Valued Functions

Abstract

Whenever $A = {\smallint _\Lambda } \oplus A(\lambda )\mu (d\lambda )$ is a decomposable operator on a direct integral $H = {\smallint _\Lambda } \oplus H(\lambda )\mu (d\lambda )$ of Hilbert spaces and f is a function analytic on a neighborhood of $\sigma (A)$, then we obtain that $f(A(\lambda ))$ is defined almost everywhere and $f(A)(\lambda ) = f(A(\lambda ))$ almost everywhere. This relationship is used to study operators A, on a separable Hilbert space, for which some analytic function A is a normal operator. Two main results are obtained. Let f be an analytic function on a neighborhood of the spectrum of an operator A. If $f''(z) \ne 0$ for all z in the spectrum of A and if $f(A)$ is a normal operator, then A is similar to a binormal operator. It is known that a binormal operator is unitarily equivalent to the direct sum of a normal and a two by two matrix of commuting normal operators. As above if $f(A)$ is normal and in addition, $f(z) - {\zeta _0}$ has at most two roots counted to their multiplicity for each ${\zeta _0}$ in the spectrum of N, then A is a binormal operator.