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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 \(R(\sigma(T))\neq C(\sigma (T))\), 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.

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.

Theorem 4. If T is a hyponormal operator acting on a given infinite dimensional, separable, complex Hilbert space, and \(R(\sigma(T))\neq C(\sigma (T))\), 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.

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