On a functional calculus for decomposable operators and applications to normal, operator-valued functions

Abstract

Whenever = J. © A(\)p(d\) is a decomposable operator on a direct integral // = f. © H(\)p(d\) of Hilbert spaces and / is a function analytic on a neighborhood of o-(A), then we obtain that f{A(\)) is defined almost every- where and f(A)(X) = f(A(\)) almost everywhere. This relationship is used to study operators A, on a separable Hilbert space, for which some analytic func- tion is a normal operator. Two main results are obtained. Let / be an ana- lytic function on a neighborhood of the spectrum of an operator A. If / (z) 4 0 for all z in the spectrum of and if f(A) is a normal operator, then 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 opera- tors. As above if f(A) is normal and in addition, /(z) — £n has at most two roots counted to their multiplicity for each T in the spectrum of N, then is a binor- mal operator. In this paper, all spaces are complex separable Hilbert spaces and all opera- tors on them are bounded. Various authors have investigated the structure of whenever a particular function of is normal. A. Brown (3) studied binormal oper- ators which included operators satisfying = XI. He gave a structure theorem for binormal operators; these results have more recently been investigated in (2), and (15). J. Stampfli has shown that if A is normal and is invertible, then is similar to a normal operator; subsequently, S. Foguel and C. Apostol have considerably generalized this result ((l), (lO), (12), (17)). In this paper, we generalize these results and show that it is possible to approach these and simi- lar problems through reduction theory of operators. Questions, concerning the struc- ture of operators, such as those investigated above will be reduced to questions about algebraic operators, which can then be handled with more elementary opera- tor theory methods. This approach seems new and essentially different from pre- vious works.