Ideals and Commutators of Operators

<p>Subideals. A subideal of operators is an ideal of J (called a J-ideal) for J an arbitraryideal of B(H). Necessary and sufficient conditions are determined for a finitely generatedsubideal to be also an ideal of B(H) and then these conditions are exploited to characterizeall finitely generated subideals. This generalizes to arbitrary ideals the 1983 work of Fongand Radjavi who determined which principal (i.e., singly generated) ideals of the ideal ofcompact operators are also ideals of B(H). Then a necessary and sufficient condition is foundfor a countably generated subideal and a subideal generated by sets of cardinality strictly lessthan the continuum to be also an ideal of B(H). This is based on the Hamel dimension of arelated quotient space. Then this condition is used to characterize these subideals and settleadditional general questions about subideals. The condition is a generalization of the notionof soft-edged ideals discovered in 2007 by Kaftal and Weiss. Examples of subideals revealsome striking differences between subideals and ideals of B(H). Also this work intersectsthe study of subideals with the study of elementary operators with coefficient constraints. </p><p>Commutators of compact operators. The 1971 commutator problem asked by Pearcyand Topping is investigated: Is every compact operator a single commutator of compactoperators? And a 1976 test question arising from work of Pearcy, Topping, Anderson andWeiss: Are any strictly positive compact operators a single commutator of compact operators?An affirmative answer to this test question for a whole class of strictly positivecompact operators is obtained. Restricted diagonalization of compact normal operators. Every normal operator in F(H)is diagonalizable by a unitary operator of the form 1 + A for 1 the identity operator andA in F(H), as is elementary to show. But in an operator ideal I properly containing F(H) we found normalcompact operators that are not diagonalizable by unitary operators of the form 1 + A forA in I. Indeed, a necessary condition that a normal operator X in I is diagonalizable by aunitary operator of the form 1+A with A in I is: X-D in I^2 where D is a diagonal operatorunitarily equivalent to X. In addition, the spectral characterization of those A (compact ornot) for which 1 + A is a unitary operator is obtained, that is, 1 + A is unitary if and onlyif A is a normal operator and its spectrum is contained in the circle -1 + T where T is theunit circle. This group of unitary operators of the form 1 + A for A in I is used to obtainuncountably many conjugacy classes of Cartan subalgebras of I.</p>

Name thesis:degree_name

PhD

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Arts and Sciences: Mathematical Sciences

Grantor dc:publisher

University of Cincinnati

Year dc:date

2012

Author dc:creator

• Patnaik, Sasmita

Contributors dc:contributor

• Weiss, Gary

• Theoretical Mathematics
• Ideals
• Operator ideals
• Principal ideals
• Subideals
• Lattices

Statement dc:rights

• unrestricted
• This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws.

Language dc:language

English

Repository record dc:identifier

http://rave.ohiolink.edu/etdc/view?acc_num=ucin1353343026

OAI identifier oai:identifier

oai:etd.ohiolink.edu:ucin1353343026