**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:l*_{1}->l_{1} is the operator without a
non-trivial closed invariant subspace constructed by C.J.Read, then
there are three operators *S*_{1}, *S*_{2}
and *K* (non-multiples of the identity) such that *T*
commutes with *S*_{1}, *S*_{1} commutes with
*S*_{2}, *S*_{2} 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"*.

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Last modified: Sun Sep 5 00:32:56 MDT 2004