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 Foiaş et al. in C. Foiaş, I.B. Jung, E. Ko, C. Pearcy, $\textit{J. Funct. Anal. } \textbf{253}(2007), 628-646,$ C. Foiaş, I.B. Jung, E. Ko, C. Pearcy, $\textit{Indiana Univ. Math. J. } \textbf{57}(2008), 2745-2760$, C. Foiaş, 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 $K$.

DOI: http://dx.doi.org/10.7900/jot.2020feb17.2269
Keywords: scalar-type spectral operators, decomposable operators, compact perturbations, hyperinvariant subspaces

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