On spectral properties of positive operators

Citation

Zhang, Xiao-Dong (1991) On spectral properties of positive operators. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/r9x6-7m39. https://resolver.caltech.edu/CaltechTHESIS:04112011-131850601

Abstract

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.

Thesis Files