Functional calculus and duality for closed operators (Q1071282)

This paper develops a functional calculus of closed operators which admit a general type of spectral decomposition of the underlying Banach space. It is shown that the spectral decomposition property (SDP) of such a linear operator T is inherited by f(T), f being the homomorphism of the functional calculus. Conversely, if the function f is nonconstant on every component of its domain which intersects the spectrum of T, then f(T) decomposable (in the sense of Foiaş) implies that T has the SDP. A spectral duality theorem follows as a corollary. The paper concludes with an analytic type property for the canonical embedding J of the underlying Banach space X into its second dual \(X^{**}\).