Compact perturbations of scalar type spectral operators

Abstract

We consider compact perturbations S = D Λ + K of normal diagonal operators D Λ by certain compact operators. Sufficient conditions for K to ensure the existence of non-trivial hyperinvariant subspaces for S have recently been obtained by Foia\c{s} et al. in C.\ Foia\c{s}, I.B.\ Jung, E.\ Ko, C. Pearcy, \textit{J.\ Funct. Anal.} \textbf{253}(2007), 628--646, C.\ Foia\c{s}, I.B.\ Jung, E.\ Ko, C.~Pearcy, \textit{Indiana Univ.\ Math.\ J.} \textbf{57}(2008), 2745--2760, {C.\ Foia\c{s}, I.B.\ Jung, E.\ Ko, C.Pearcy}, \textit{J.\ Math.\ Anal.\ Appl.} \textbf{375}(2011), 602--609 (followed by Fang--Xia \textit{J.\ Funct. Anal} \textbf{263}(2012), 135-1377, and Klaja \textit{J.\ Operator Theory} \textbf{73}(2015), 127--142, by using certain spectral integrals along straight lines through the spectrum of S . In this note, the authors use circular cuts and get positive results under less restrictive local conditions for K .