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

We study the local functional calculus of an operator T having the singlevalued 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 singlevalued 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.