Functional calculus and duality for closed operators |
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Authors: | I Erdelyi Wang Shengwang |
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Affiliation: | Temple University, Philadelphia, Pennsylvania 19122 USA;Nanjing University, Nanjing, The People''s Republic of China |
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Abstract: | In this paper, we continue our spectral-theoretic study [8] of unbounded closed operators in the framework of the spectral decomposition property and decomposable operators. Given a closed operator T with nonempty resolvent set, let f → f(T) be the homomorphism of the functional calculus. We show that if T has the spectral decomposition property, then f(T) is decomposable. Conversely, if f is nonconstant on every component of its domain which intersects the spectrum of T, then f(T) decomposable implies that T has the spectral decomposition property. A spectral duality theorems follows as a corollary. Furthermore, we obtain an analytic-type property for the canonical embedding J of the underlying Banach space X into its second dual . |
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