On subnormal operators

Abstract

Let T be the adjoint of a subnormal operator defined on a Hilbert space H. For any closed set δ \delta , let X T ( δ ) = { x ∈ H {X_T}(\delta ) = \{ x \in H : there exists an analytic function f x : C ∖ δ → H {f_x}:{\text {C}}\backslash \delta \to H such that ( z − T ) f x ( z ) ≡ x } (z - T){f_x}(z) \equiv x\} . It is shown that T is decomposable (resp. normal) if X T ( ∂ G α ) {X_T}(\partial {G_\alpha }) is closed (resp. if X T ( ∂ G α ) = { 0 } ) {X_T}(\partial {G_\alpha }) = \{ 0\} ) for a certain family { G α } \{ {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.