Invariant subspaces for subscalar operators
One of the famous results of \textit{S. Brown} [Ann. Math. II. Ser. 125, 93- 103 (1987; Zbl 0635.47020)] says that on a Hilbert space each hyponormal operator with thick spectrum has a non-trivial invariant subspace. The proof uses a result of M. Putinar which states that each hyponormal operator is subscalar, i.e. the restriction of an operator generalized scalar in the sense of C. Foiaş. In an attempt to generalize S. Brown's result to the context of arbitrary Banach spaces \textit{E. Albrecht} and \textit{B. Chevreau} proved that each restriction or quotient of a decomposable operator with sufficiently rich spectrum wich acts on a quotient of closed subspaces of \(\ell\) p \((1<p<\infty)\) has non-trivial invariant subspace [J. Oper. Theory 18, 339-372 (1987)]. In the paper under review the condition to make the Scott Brown technique work is not demanded for the space, but for the operator. The main result states that on an arbitrary Banach space each subscalar operator A with Int(\(\sigma\) (A))\(\neq \emptyset\) has a non-trivial, even rationally invariant subspace. The proof uses the Scott Brown technique together with some ideas of C. Apostol.
restriction or quotient of a decomposable operator
scientific article; zbMATH DE number 4061961
scientific article