A Local Functional Calculus and Related Results on the Single-Valued Extension Property

Abstract

We study the local functional calculus of an operator T having the single-valued extension property. We consider a vector f(T, v) for an analytic function f on a neighborhood of the local spectrum of a vector v with respect to T and show that the local spectrum of v and the local spectrum of f(T, v)are equal with the possible exception of points of the local spectrum of v that are zeros of f, that is, we show that \( \sigma_{T} \)(v) is equal to \( \sigma_{T}\)(f(T,v)) union the set of zeros of f on \( \sigma_{T} \)(v). This local functional calculus extends the Riesz functional calculus for operators. For an analytic function f on a neighborhood of \( \sigma\)(T), we use the above mentioned proposition to obtain proofs of the results that if T has the single-valued extension property, then f(T) also has the single-valued extension property, and conversely if f is not constant on each connected component of a neighborhood of \( \sigma\)(T) and f(T) has the singlevalued extension property, then T also does.