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On the Uniqueness of Wave Operators Associated with Non-Trace Class Perturbations

Published online by Cambridge University Press:  20 November 2018

Jingbo Xia
Jingbo Xia*
Affiliation:
Department of Mathematics State University of New York at Buffalo Buffalo, New York 14260 USA, e-mail: jxia@acsu.buffalo.edu
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Abstract

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Voiculescu has previously established the uniqueness of the wave operator for the problem of ${{\mathcal{C}}^{(0)}}$-perturbation of commuting tuples of self-adjoint operators in the case where the norm ideal $\mathcal{C}$ has the property ${{\lim }_{n\,\to \,\infty }}\,{{n}^{-1/2}}\,\left\| {{P}_{n}} \right\|\mathcal{C}\,=\,0$, where $\{{{P}_{n}}\}$ is any sequence of orthogonal projections with rank$({{P}_{n}})\,=\,n$. We prove that the same uniqueness result holds true so long as $\mathcal{C}$ is not the trace class. (It is well known that there is no such uniqueness in the case of trace-class perturbation.)

Keywords

47A4047B10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 47 , Issue 1 , 01 March 2004 , pp. 144 - 151
Copyright
Copyright © Canadian Mathematical Society 2004

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