K-spectral sets and invariant subspaces

Abstract

In this paper we show that given any Banach space operator T whose spectrum is a k-spectral set, its adjoint T* has a nontrivial invariant subspace. INTRODUCTIONThe dual algebras techniques, initiated by S. Brown in [5], have been very successful in dealing with Hilbert space operators. Extending these techniques to the setting of Banach space operators is a rather difficult task. However, in recent years, several important results in this direction have been obtained (see [2], [31, [111, [121, [131 and [14]) and there are still many open problems in this area.In this note we show that some of the results relating k-spectral sets and invariant subspaces for operators on Hilbert spaces (see [1], [4], [17] and [18]) can be extended to operators on arbitrary Banach spaces.