Invariant subspace problem and spectral properties of bounded linear operators on Banach spaces, Banach lattices, and topological vector spaces, Ph.D. thesis, defended in April 1999 at the Department of Mathematics at the University of Illinois at Urbana-Champaign.

Advisors: Yuri Abramovich and Peter Loeb.

Abstract

Chapter 1 deals with the Invariant Subspace Problem for Banach spaces and Banach lattices. First, we show that the celebrated Lomonosov theorem cannot be improved by increasing the number of commuting operators. Specifically, we prove that if T:l1->l1 is the operator without a non-trivial closed invariant subspace constructed by C.J.Read, then there are three operators S1, S2 and K (non-multiples of the identity) such that T commutes with S1, S1 commutes with S2, S2 commutes with K, and K is compact. We also show that the commutant of T contains only series of T. Further, we show that the modulus of the quasinilpotent operator without an invariant subspace constructed by C.J.Read in 1997 has an invariant subspace (and even an eigenvector). This answers a question posed by Y.Abramovich, C.Aliprantis and O.Burkinshaw.

In Chapter 2 we develop a version of spectral theory for bounded linear operators on topological vector spaces. We show that the Gelfand formula for spectral radius and Neumann series can still be naturally interpreted for operators on topological vector spaces. Of course, the resulting theory has many similarities to the conventional spectral theory of bounded operators on Banach spaces, though there are several important differences. The main difference is that an operator on a topological vector space has several spectra and several spectral radii, which fit a well-organized pattern.

In Chapter 3 we use the results of Chapter 2 to prove locally-convex versions of some results on the Invariant Subspace Problem on Banach lattices obtained by Y.Abramovich, C.Aliprantos, and O.Burkinshaw in 1993-98. For example, we show that if S and T are two commuting positive continuous operators with finite spectral radii on a locally convex-solid vector lattice, T is locally quasinilpotent at a positive vector, and S dominates a positive compact operator, then S and T have a common closed non-trivial invariant subspace.

The results of Chapter 1 were published in "Lomonosov's theorem cannot be extended to chains of four operators" and in "On the modulus of C.J. Read's operator". The results of Chapter 2 were published in "Spectral radii of bounded operators on topological vector spaces".

Back to the list of papers
Last modified: Sun Sep 5 00:32:56 MDT 2004