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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A trace formula for isometric pairs
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by Rongwei Yang PDF
Proc. Amer. Math. Soc. 131 (2003), 533-541 Request permission


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:
  • 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:
  • MathSciNet review: 1933344