On subnormal operators

Radjabalipour, Mehdi

doi:10.1090/S0002-9947-1975-0377574-2

On subnormal operators
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Trans. Amer. Math. Soc. 211 (1975), 377-389 Request permission

Abstract:

Let T be the adjoint of a subnormal operator defined on a Hilbert space H. For any closed set $\delta$, let ${X_T}(\delta ) = \{ x \in H$: there exists an analytic function ${f_x}:{\text {C}}\backslash \delta \to H$ such that $(z - T){f_x}(z) \equiv x\}$. It is shown that T is decomposable (resp. normal) if ${X_T}(\partial {G_\alpha })$ is closed (resp. if ${X_T}(\partial {G_\alpha }) = \{ 0\} )$ for a certain family $\{ {G_\alpha }\}$ of open sets. Some of the results are extended to the case that T is the adjoint of the restriction of a spectral or decomposable operator to an invariant subspace.

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