Decomposability and triangularizability of positive operators on Banach lattices

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منو

" Decomposability and triangularizability of positive operators on Banach lattices "
M. T. Jahandideh H. Radjavi

Document Type	:	Latin Dissertation
Language of Document	:	English
Record Number	:	1113137
Doc. No	:	TLpq304380235
Main Entry	:	H. Radjavi
	:	M. T. Jahandideh
Title & Author	:	Decomposability and triangularizability of positive operators on Banach lattices\ M. T. JahandidehH. Radjavi
College	:	Dalhousie University (Canada)
Date	:	1997
student score	:	1997
Degree	:	Ph.D.
Page No	:	108
Abstract	:	The main results in this thesis are about invariant subspaces of multiplicative semigroups of quasinilpotent positive operators on Banach lattices. There are some known results that guarantee the existence of a non-trivial closed invariant ideal for a quasinilpotent positive operator on certain spaces, for example on usdC\sb0(\Omega)usd with usd\Omegausd a locally compact Hausdorff space or on a Banach lattice with atoms. Some recent results also guarantee the existence of non-trivial closed invariant ideals for a compact quasinilpotent positive operator on an arbitrary Banach lattice. In fact it is known that given such an operator T, on a real or complex Banach lattice, there is a nontrivial closed ideal which is invariant under all positive operators that commute with T. This thesis deals with invariant ideals for families of positive operators on Banach lattices. In particular it studies ideal-decomposable and ideal-triangularizable semi-groups of positive operators. We show that in certain Banach lattices compactness is not required for the existence of hyperinvariant closed ideals for a quasinilpotent positive operator. We also show that in those Banach lattices a semigroup of quasinilpotent positive operators might be decomposable without imposing any compactness condition. We generalize the fact that the only irreducible usdC\sb{p}usd-closed subalgebra of usdC\sb{p}usd is usdC\sb{p}usd itself to extend some recent reducibility results and apply them to derive some decomposability theorems concerning a collection of quasinilpotent positive operators on reflexive Banach lattices. We use these results for "ideal-triangularization", i.e., we construct a maximal closed ideal chain, each of whose members is invariant under a certain collection of operators that are related to compact positive operators, or to quasinilpotent positive operators.
Subject	:	Mathematics
	:	Pure sciences