Journal of Operator Theory

Volume 47, Issue 2, Spring 2002 pp. 219-243.

An integral representation for semigroups of unbounded normal operators

Authors: P. Ressel (1), and W.J. Ricker (2)
Author institution: (1) Math.-Geogr. Fakultat, Katholische Universitaet Eichst, D--85071 Eichstatt, Germany
(2) School of Mathematics, University of New South Wales, Sydney, N.S.W.\ 2052, Australia

Summary: An integral representation for semigroups $\{ U_s \} _{s \in S }$ of unbounded normal operators in a Hilbert space $ H $ is presented which admits a significantly larger class of semigroups $ S $ than usual. In particular, $ S $ need not have a topology and so the traditional assumption that the functions $ s \lm \langle U_s x, y \rangle$, for suitable elements $ x, y \in H$, are continuous is no longer a requirement. The classical spectral theorem for a single (unbounded) normal or selfadjoint operator is a {\it consequence} of the main result; the point is that the techniques used do not rely on the fact that a normal operator has a spectral decomposition via its resolution of the identity.

Keywords: Semigroup representation, positive definiteness, normal operator, spectral theorem

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