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Hyponormal operators with thick spectra have invariant subspaces. (English) Zbl 0635.47020
The following main theorem is proved: Theorem 4. If T is a hyponormal operator acting on a given infinite dimensional, separable, complex Hilbert space, and $R(\sigma(T))\ne C(\sigma (T))$, then T has an invariant subspace where $C(\sigma(T))$ denotes the space of continuous functions on $\Sigma(T)$, the spectrum of T, and $R(\sigma(T))$ denotes the closure in $C(\sigma(T))$ of the rational functions with poles off $\sigma$ (T). The invariant subspace proof for subnormal operators obtained by the author [Integral Equations Oper. Theory 1, 310-333 (1978; Zbl 0416.47009)] does not adapt to hyponormal situation. The result stated above is reformulated, in the equivalent result that follows: If T is a hyponormal operator and G is a non-empty open subset of the complex plane such that $\sigma(T)\cap G$ is dominating in G, then T has a nontrivial invariant subspace. {\it C. Apostol} [Rev. Roumaine Math. Pures Appl. 13, 1481-1528 (1968; Zbl 0176.437)] proved the equivalence of these statements for subnormal operators. The present paper used the techniques of Apostol and makes them work for the more general operators on a Hilbert space that are decomposable but not unconditionally decomposable, and in the process the invariant subspace result for hyponormal operators is achieved.
Reviewer: R.K.Bose

47B20Subnormal operators, hyponormal operators, etc.
47A15Invariant subspaces of linear operators
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