## 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.
Reviewer: R.K.Bose

### MSC:

 47B20 Subnormal operators, hyponormal operators, etc. 47A15 Invariant subspaces of linear operators

### Citations:

Zbl 0416.47009; Zbl 0176.437
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