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Journal of Operator Theory

Volume 86, Issue 1, Summer 2021  pp. 163-188.

Compact perturbations of scalar type spectral operators

Authors:  Ernst Albrecht (1), Bernard Chevreau (2)
Author institution: (1) Fachrichtung 6.1 - Mathematik, Universitaet des Saarlandes, 66041 Saarbruecken, Germany
(2) Institut de Mathematiques de Bordeaux, Universite de Bordeaux, 351, cours de la Liberation, F 33 405 Talence Cedex, France

Summary: We consider compact perturbations $S=D_\Lambda+K$ of normal diagonal operators $D_\Lambda$ 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 $K$.

Keywords: scalar-type spectral operators, decomposable operators, compact perturbations, hyperinvariant subspaces

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