Title: COMPACT PERTURBATIONS OF NORMAL OPERATORS: WHERE ARE THEIR INVARIANT SUBSPACES?

Abstract: In this talk, we will address the problem regarding the existence of non-trivial closed invariant subspaces of compact perturbations of normal operators acting boundedly on separable, infinite-dimensional complex Hilbert spaces. After considering the finite-rank case, we will show that a large class of such operators are decomposable, extending, in particular, recent results of Foias, Jung, Ko and Pearcy.

Decomposable operators were introduced by Foias in the sixties. Many operators acting on Hilbert spaces are decomposable: unitary operators, self-adjoint operators, or more generally, normal operators. In a broad sense, decomposable operators have the most general kind of spectral decomposition possible. Consequently, every operator in the aforementioned class has a rich spectral structure and plenty of non-trivial closed invariant subspaces.

Based on joint works with F. J. González-Doña.

In conjunction with this colloquium, Professor Gallardo-Gutierrez is giving a special lecture: THE  INVARIANT  SUBSPACE  PROBLEM: GENERAL OPERATOR THEORY  VS.  CONCRETE  OPERATOR  THEORY?, at 12:30–1:20 p.m. in the same day of this colloquium at Muhly Lounge, MLH on introductory materials related with this colloquium.

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