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 $K$.
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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