## A trace formula for isometric pairs

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- by Rongwei Yang
- Proc. Amer. Math. Soc.
**131**(2003), 533-541 - DOI: https://doi.org/10.1090/S0002-9939-02-06687-X
- Published electronically: June 5, 2002
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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 \[ tr[V_{1}^{*},V_{1},V_{2}^{*},V_{2}]=-2ind(V_{1}, V_{2}),\] 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.

## References

- O. P. Agrawal, D. N. Clark, and R. G. Douglas,
*Invariant subspaces in the polydisk*, Pacific J. Math.**121**(1986), no. 1, 1–11. MR**815027** - M. F. Atiyah,
*$K$-theory*, W. A. Benjamin, Inc., New York-Amsterdam, 1967. Lecture notes by D. W. Anderson. MR**0224083** - C. A. Berger, L. A. Coburn, and A. Lebow,
*Representation and index theory for $C^*$-algebras generated by commuting isometries*, J. Functional Analysis**27**(1978), no. 1, 51–99. MR**0467392**, DOI 10.1016/0022-1236(78)90019-8 - L. G. Brown, R. G. Douglas, and P. A. Fillmore,
*Extensions of $C^*$-algebras and $K$-homology*, Ann. of Math. (2)**105**(1977), no. 2, 265–324. MR**458196**, DOI 10.2307/1970999 - R. W. Carey and J. D. Pincus,
*On local index and the cocycle property for Lefschetz numbers*, Topics in operator theory and interpolation, Oper. Theory Adv. Appl., vol. 29, Birkhäuser, Basel, 1988, pp. 45–86. MR**945003** - Raul E. Curto,
*Fredholm and invertible $n$-tuples of operators. The deformation problem*, Trans. Amer. Math. Soc.**266**(1981), no. 1, 129–159. MR**613789**, DOI 10.1090/S0002-9947-1981-0613789-6 - R. G. Douglas and C. Foias,
*A classification of multi-isometries*, preprint. - Ronald G. Douglas,
*Banach algebra techniques in operator theory*, 2nd ed., Graduate Texts in Mathematics, vol. 179, Springer-Verlag, New York, 1998. MR**1634900**, DOI 10.1007/978-1-4612-1656-8 - R. G. Douglas and Dan Voiculescu,
*On the smoothness of sphere extensions*, J. Operator Theory**6**(1981), no. 1, 103–111. MR**637004** - P. Ghatage and V. Mandrekar,
*On Beurling type invariant subspaces of $L^2(\textbf {T}^2)$ and their equivalence*, J. Operator Theory**20**(1988), no. 1, 83–89. MR**972182** - D. Gasper and N. Suciu,
*Intertwining properties of isometric semigroups and Wold type decompositions*, Operator Theory: Adv. and Appl., 24 (1987), 183-193. - J. William Helton and Roger E. Howe,
*Traces of commutators of integral operators*, Acta Math.**135**(1975), no. 3-4, 271–305. MR**438188**, DOI 10.1007/BF02392022 - L. Hörmander,
*The Weyl calculus of pseudodifferential operators*, Comm. Pure Appl. Math.**32**(1979), no. 3, 360–444. MR**517939**, DOI 10.1002/cpa.3160320304 - I. Suciu,
*On the semi-groups of isometries*, Studia Math.**30**(1968), 101–110. MR**229093**, DOI 10.4064/sm-30-1-101-110 - Rongwei Yang,
*BCL index and Fredholm tuples*, Proc. Amer. Math. Soc.**127**(1999), no. 8, 2385–2393. MR**1605949**, DOI 10.1090/S0002-9939-99-04895-9 - Rongwei Yang,
*The Berger-Shaw theorem in the Hardy module over the bidisk*, J. Operator Theory**42**(1999), no. 2, 379–404. MR**1717024** - R. Yang,
*Operator theory in the Hardy space over the bidisk (III)*, J. of Funct. Anal., 186 (2001), 521–545. - R. Yang,
*Beurling’s phenomenon in two variables*, preprint.

## Bibliographic 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