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The generalized Wold decomposition for subnormal operators. (English) Zbl 0645.47021
Let S be a subnormal operator on a Hilbert space and N its minimal normal extension. If \(\Omega\) is a finitely connected domain bounded by Jordan curves such that the spectrum of S is contained in \(\Omega\) and the spectrum of N is contained in \(\partial \Omega\), M. B. Abrahamse and R. G. Douglas [Adv. Math. 19, 106-148 (1976; Zbl 0321.47019)] have shown that S can be represented as the direct sum of a normal operator and a bundle shift over \(\Omega\) (i.e. an operator \(f\mapsto zf\) for \(f\in H^ 2[E]\) where E is a certain flat unitary bundle over \(\Omega\) and \(H^ 2[E]\) consists of those analytic sections \(\phi\) of E for which the functions \(z\mapsto \| \phi (z)\|^ 2\) have harmonic majorants on \(\Omega)\). In the present paper this Wold decomposition is extended to the case of subnormal operators with less restrictive conditions on \(\Omega\). For the proof a result about the absolute continuity of the spectral measure of N with respect to the harmonic measure on \(\partial \Omega\) and a Beurling-type description of invariant subspaces are obtained.
Reviewer: G.Garske

47B20 Subnormal operators, hyponormal operators, etc.
47A10 Spectrum, resolvent
47A15 Invariant subspaces of linear operators
Full Text: DOI
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