Journal of Operator Theory

Volume 63, Issue 1, Winter 2010 pp. 101-114.

Reflexivity and hyperreflexivity of the space of locally intertwining operators

Authors: Janko Bracic, Vladimir Mueller (2), and Michal Zajac (3)
Author institution: (1) University of Ljubljana, IMFM, Jadranska ul. 19, SI-1000 Ljubljana, Slovenia
(2) Mathematical Institute, Czech Academy of Sciences, Zitna 25, 115 67 Prague 1, Czech Republic
(3) Department of Mathematics, Slovak University of Technology, SK-812 19 Bratislava, Slovakia

Summary: An operator $S$ is a local intertwiner of operators $A$ and $B$ at vector $e$ if $SAe=BSe$. We characterize the spaces of all local intertwiners $\Int(A,B;e)$ that are reflexive (hyperreflexive). We show that in all interesting cases the reflexivity (hyperreflexivity) of $\Int(A,B;e)$ depends only on $B$ and is independent of $A$ and $e$. This has consequences concerning the reflexivity of the space of intertwiners $\Int(A,B)$ and of the commutant of an operator.

Keywords: Space of intertwiners, space of local intertwiners, commutant, local commutant, reflexivity, hyperreflexivity

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