Title: Spectral Representations for Some Unbounded Normal Operators
Abstract: Introduction.In a recent paper [17] we obtained a spectral representation for the bounded normal operator solutions of a certain functional equation whose special cases include the semigroup equation (UßV = UßUv) as well as many others.In this note we continue our study of the abstract functional equation expressed in terms of algebras of measures.We shall consider therefore a mapping z -» Uz of a locally compact space Z into the set of unbounded normal operators on a Hilbert space.Since the operators are not necessarily bounded, the conditions imposed on the mapping z -» Uz vary somewhat from the conditions in the bounded case.The main result (Theorem 1) which we obtain here is a spectral representation analogous to the result valid in the bounded case [17; 18, Theorem 2, p. 8].The methods which we use are suggested by the papers [12] and [13] of C. Ionescu Tulcea.For the general theory and terminology of unbounded operators we refer the reader to the works of E. Hille and R. Phillips [10], C. Ionescu Tulcea [11], F. Riesz and B. Sz.-Nagy [21].Spectral representations for semigroups of unbounded operators are also discussed in A. Devinatz [3], A. Devinatz and A. E. Nussbaum [4; 5], R. Getoor [7], C. Ionescu Tulcea and A. Simon [14] and A .E. Nussbaum [19].Theorem 1 of this paper contains the main part of Theorem 1 in [14] (and henceforth certain of the results given in [3; 4; 5; 7; 19]).The setting of the problem in algebras of measures is given in § §1, 2, and 5. Spectral families of Radon measures and direct sums thereof are summarized in § §3 and 4. The main result is stated and proved in §6.In §8 we apply the main theorem to derive a corollary which in the bounded case extends a result of S. Kurepa [16].1. Algebras of measures.We denote by Z a locally compact space and by K(Z) the vector space of continuous complex-valued functions / defined on Z and having compact support.For any compact set AcZ let M(Z,A) be the Banach space (endowed with the norm p -> I p | = sup {| p(f) | : fe K(Z), ||/|| " i£ 1}) of all measures in M(Z) having support contained in A. We shall consider M(Z)