Spectral properties between operator matrices and Helton class

Abstract

Abstract. In this paper, we study properties of Helton class of an operator matrix. In particular, we show that some upper operator matrix belongs to Helton class of an operator matrix have the property (β ) ε . As an application, we get that such operators have nontrivial invariant subspaces. Finally, we prove that Helton class preserves the generalized Weyl's theorem under some conditions.Mathematics subject classification (2010): Primary 47A11, 47A53; secondary 47A10. Keywords and phrases: Helton class, operator matrix, the property (β ) ε , the generalized Weyl's theorem. R E F E R E N C E