On the relation between an operator and its self-commutator

TY - JOUR

T1 - On the relation between an operator and its self-commutator

AU - Filonov, N.

AU - Safarov, Y.

PY - 2011/5/15

Y1 - 2011/5/15

N2 - We show that a bounded operator A on a Hilbert space belongs to a certain set associated with its self-commutator [A*, A], provided that A - zI can be approximated by invertible operators for all complex numbers z. The theorem remains valid in a general C*-algebra of real rank zero under the assumption that A - zI belong to the closure of the connected component of unity in the set of invertible elements. This result implies the Brown-Douglas-Fillmore theorem and Huaxin Lin's theorem on almost commuting matrices. Moreover, it allows us to refine the former and to extend the latter to operators of infinite rank and other norms (including the Schatten norms on the space of matrices). The proof is based on an abstract theorem, which states that a normal element of a C*-algebra of real rank zero satisfying the above condition has a resolution of the identity associated with any open cover of its spectrum. Crown Copyright (C) 2011 Published by Elsevier Inc. All rights reserved.

AB - We show that a bounded operator A on a Hilbert space belongs to a certain set associated with its self-commutator [A*, A], provided that A - zI can be approximated by invertible operators for all complex numbers z. The theorem remains valid in a general C*-algebra of real rank zero under the assumption that A - zI belong to the closure of the connected component of unity in the set of invertible elements. This result implies the Brown-Douglas-Fillmore theorem and Huaxin Lin's theorem on almost commuting matrices. Moreover, it allows us to refine the former and to extend the latter to operators of infinite rank and other norms (including the Schatten norms on the space of matrices). The proof is based on an abstract theorem, which states that a normal element of a C*-algebra of real rank zero satisfying the above condition has a resolution of the identity associated with any open cover of its spectrum. Crown Copyright (C) 2011 Published by Elsevier Inc. All rights reserved.

U2 - 10.1016/j.jfa.2011.02.011

DO - 10.1016/j.jfa.2011.02.011

M3 - Article

VL - 260

SP - 2902

EP - 2932

JO - JOURNAL OF FUNCTIONAL ANALYSIS

JF - JOURNAL OF FUNCTIONAL ANALYSIS

IS - 10

ER -