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"Operator ideals on ordered Banach spaces" Open Access


Other title
domination problem
operator ideal
strictly singular operator
innesential operator
noncommutative function space
operator space
ordered Banach space
Type of item
Degree grantor
University of Alberta
Author or creator
Spinu, Eugeniu
Supervisor and department
Tcaciuc, Adi ( Grant McEwan University)
Troitsky, Vladimir G. (Department of Mathematical and Statistical Sciences)
Examining committee member and department
Litvak, Alexander (Department of Mathematical and Statistical Sciences)
Yaskin, Vlad (Department of Mathematical and Statistical Sciences)
Hillen, Thomas (Department of Mathematical and Statistical Sciences)
Troitsky, Vladimir G. (Department of Mathematical and Statistical Sciences)
Tcaciuc, Adi ( Grant McEwan University)
Runde, Volker (Department of Mathematical and Statistical Sciences)
Department of Mathematical and Statistical Sciences
Date accepted
Graduation date
Doctor of Philosophy
Degree level
In this thesis we study operator ideals on ordered Banach spaces such as Banach lattices, $C^*$-algebras, and noncommutative function spaces. The first part of this work is concerned with the domination problem: the relationship between order and algebraic ideals of operators. Fremlin, Dodds and Wickstead described all Banach lattices on which every operator dominated by a compact operator is always compact. First, we show that even if the dominated operator is not compact it still belongs to a relatively small class of operators, namely, the ideal of inessential operators. A similar question is studied for strictly singular operators. In particular, we show that the cube of every operator, dominated by a strictly singular operator, is inessential. Then we provide a complete solution of the domination problem for compact and weakly compact operators acting between $C^{*}$-algebras and noncommutative function spaces. Finally, we consider the domination problem for weakly compact operators acting on general noncommutative function spaces. The second part is devoted to the operator ideal structure of the algebra of all linear bounded operators on a Banach space. First, we investigate the existence of non-trivial proper ideals on Lorentz sequence spaces and characterize some of them. Second, we look at the coincidence of some classical operator ideals, such as of compact, strictly singular, innesential, and Dunford-Pettis operators acting on noncommutative $L_p$-spaces. In particular, we obtain a characterization of strictly singular and inessential operators acting either between discrete noncommutative $L_p$-spaces or $L_p$-spaces, associated with a hyperfinite von Neumann algebras with finite trace.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
Citation for previous publication
E. Spinu. Dominated inessential operators. J. Math. Anal. Appl., 383(2), 259--264, 2011.A. Kaminska, A. I. Popov, E. Spinu, A. Tcaciuc, V. G. Troitsky. Norm closed operator ideals in Lorentz sequence spaces. J. Math. Anal. Appl., 389(1), 247--260, 2012.T. Oikhberg, E. Spinu, Domination problem in non-commutative setting (preprint at, 2012

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