Spectral Measures, Generalized Resolvents, and Functions of Positive Type

摘要

The theme of this report is the interplay of classes of operator-valued functions (E sub t, R sub lambda, V sub s), acting in a Hilbert space. In the prototype these functions are associated with a self-adjoint T : E sub t is the spectral function of T, R sub lambda is its resolvent, and V sub s is the unitary group e to the minus is T. In another circumstance, as studied by M. A. Naimark and A. V. Strauss, E sub t is the generalized spectral function of a symmetric operator, and R sub lambda is its generalized resolvent. In still another case one identifies V sub s as a semigroup of contraction operators - here important theorems carry the names of E. Hille, K. Yosida, and B. Sz.-Nagy. It is our thesis that by systematically developing, on a suitable level of generality, the interrelations among operator-valued functions (E sub t, R sub lambda, V sub s), one finds a unified approach to various specializations, obtains new, brief, and transparent proofs of the known facts concerning them, and uncovers new relations previously unnoticed. (Author)