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A trace formula for isometric pairs

Author: Rongwei Yang
Journal: Proc. Amer. Math. Soc. 131 (2003), 533-541
MSC (2000): Primary 47A13
Published electronically: June 5, 2002
MathSciNet review: 1933344
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Abstract: It is well known that for every isometry $V$, $tr[V^{*}, V]=-ind(V).$ This fact for the shift operator is a basis for many important developments in operator theory and topology. In this paper we prove an analogous formula for a pair of isometries $(V_{1}, V_{2})$, namely

\begin{displaymath}tr[V_{1}^{*},V_{1},V_{2}^{*},V_{2}]=-2ind(V_{1}, V_{2}),\end{displaymath}

where $[V_{1}^{*},V_{1},V_{2}^{*},V_{2}]$ is the complete anti-symmetric sum and $ind(V_{1}, V_{2})$ is the Fredholm index of the pair $(V_{1}, V_{2})$. The major tool is what we call the fringe operator. Two examples are considered.

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Additional Information

Rongwei Yang
Affiliation: Department of Mathematics, Arizona State University, Tempe, Arizona 85287
Address at time of publication: Department of Mathematics and Statistics, State University of New York at Albany, Albany, New York 12222

Received by editor(s): March 20, 2001
Received by editor(s) in revised form: September 25, 2001
Published electronically: June 5, 2002
Additional Notes: The author was partially supported by a grant from the National Science Foundation (DMS 9970932)
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2002 American Mathematical Society