Journal of Operator Theory

Volume 69, Issue 1, Winter 2013 pp. 3-16.

Stable and norm-stable invariant subspaces

Authors: Alexander Borichev (1), Don Hadwin (2), Hassan Yousefi (3)
Author institution: (1) Centre de Mathematiques et Informatique, Aix-Marseille Universite, 39 rue Frederic Joliot-Curie, 13453 Marseille, France
(2) Mathematics Department, University of New Hampshire, Durham, NH 03824, U.S.A.
(3) Mathematics Department, California State University, Fullerton, CA 92831, U.S.A.

Summary: We prove that if $T$ is an operator on an infinite-dimensional Hil\-bert space whose spectrum and essential spectrum are both connected and whose \textit{Fredholm index} is only $0$ or $1$, then the only nontrivial \textit{norm-stable invariant subspaces} of $T$ are the finite-dimensional ones. We also characterize norm-stable invariant subspaces of any weighted unilateral shift operator. We show that \textit{quasianalytic shift operators} are points of norm continuity of the lattice of the invariant subspaces. We also provide a necessary condition for strongly stable invariant subspaces for certain operators.

Keywords: stable invariant subspaces, weighted shift operator

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