Phd records
https://feeds.library.caltech.edu/people/Zhang-X-D/Phd.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenWed, 31 Jan 2024 20:05:53 +0000On spectral properties of positive operators
https://resolver.caltech.edu/CaltechTHESIS:04112011-131850601
Authors: {'items': [{'id': 'Zhang-X-D', 'name': {'family': 'Zhang', 'given': 'Xiao-Dong'}, 'show_email': 'NO'}]}
Year: 1991
DOI: 10.7907/r9x6-7m39
This thesis deals with the spectral behavior of positive operators and related ones
on Banach lattices. We first study the spectral properties of those positive operators
that satisfy the so-called condition (c). A bounded linear operator T on a Banach
space is said to satisfy the condition (c) if it is invertible and if the number 0 is in
the unbounded connected component of its resolvent set p(T). By using techniques
in complex analysis and in operator theory, we prove that if T is a positive operator
satisfying the condition (c) on a Banach lattice E then there exists a positive number a
and a positive integer k such that T^k ≥ a•I, where I is the identity operator on E. As
consequences of this result, we deduce some theorems concerning the behavior of the
peripheral spectrum of positive operators satisfying the condition (c). In particular,
we prove that if T is a positive operator with its spectrum contained in the unit circle
Γ then either σ(T) = Γ or σ(T) is finite and cyclic and consists of k-th roots of unity
for some k. We also prove that under certain additional conditions a positive operator
with its spectrum contained in the unit circle will become an isometry. Another main
result of this thesis is the decomposition theorem for disjointness preserving operators.
We prove that under some natural conditions if T is a disjointness preserving operator
on an order complete Banach lattice E such that its adjoint T' is also a disjointness
preserving operator then there exists a family of T-reducing bands {E_n : ≥ 1} U
{E_∞} of E such that T|E_n has strict period n and that T|E_∞ is aperiodic. We also
prove that any disjointness preserving operator with its spectrum contained in a
sector of angle less than π can be decomposed into a sum of a central operator and a
quasi-nilpotent operator. Among other things we give some conditions under which
an operator T lies in the center of the Banach lattice. Also discussed in this thesis
are certain conditions under which a positive operator T with σ(T) = {1} is greater
than or equal to the identity operator I.
https://thesis.library.caltech.edu/id/eprint/6294