# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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

##### MSC:
 47B20 Subnormal operators, hyponormal operators, etc. 47A15 Invariant subspaces of linear operators
Full Text: