## A trace formula for isometric pairs

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- by Rongwei Yang PDF
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**131**(2003), 533-541 Request permission

## 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 \[ tr[V_{1}^{*},V_{1},V_{2}^{*},V_{2}]=-2ind(V_{1}, V_{2}),\] 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
- Email: ryang@math.la.asu.edu
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**131**(2003), 533-541 - MSC (2000): Primary 47A13
- DOI: https://doi.org/10.1090/S0002-9939-02-06687-X
- MathSciNet review: 1933344