A trace formula for isometric pairs
Author:
Rongwei Yang
Journal:
Proc. Amer. Math. Soc. 131 (2003), 533541
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 , 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 , namely
where is the complete antisymmetric sum and is the Fredholm index of the pair . The major tool is what we call the fringe operator. Two examples are considered.
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 [At]
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 [BCL]
 C. Berger, L. Coburn, A. Lebow, Representation and index theory for algebras generated by commuting isometries, J. Functional Analysis, 27(1978), No. 1. MR 57:7251
 [BDF]
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 [CP]
 R. W. Carey and J. D. Pincus, On local index and the cocycle property for Lefschetz numbers, Oper. Theory: Adv. Appl., 29 (1988), 4586. MR 89m:47007
 [Cu]
 R. E. Curto, Fredholm and invertible ntuples of operators. The deformation problem, Trans. A.M.S., 266 (1981), 129159. MR 82g:47010
 [DF]
 R. G. Douglas and C. Foias, A classification of multiisometries, preprint.
 [Do]
 R. G. Douglas, Banach algebra techniques in operator theory, Second edition, Graduate Texts in Mathematics, 179, SpringerVerlag, New York, 1998. MR 99c:47001
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 R. G. Douglas and D. Voiculescu, On the smoothness of sphere extentions, J. Oper. Theory, 6 (1981), 103111. MR 83h:46080
 [GM]
 P. Ghatage and V. Mandrekar, On Beurling type invariant subspaces of and their equivalence, J. Operator Theory, 20 (1988), No. 1, 8389. MR 90i:47005
 [GS]
 D. Gasper and N. Suciu, Intertwining properties of isometric semigroups and Wold type decompositions, Operator Theory: Adv. and Appl., 24 (1987), 183193.
 [HH]
 J. W. Helton and R. Howe, Traces of commutators of integral operators, Acta. Math., 135 (1975), No. 34, 271305. MR 55:11106
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 L. Hörmander, The Weyl calculus of pseudodifferential operators, Acta. Math., 32 (1979), 359443. MR 80j:47060
 [Su]
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 R. Yang, BCL index and Fredholm tuples, Proc. A.M.S, Vol. 127 (1999), No. 8, 23852393. MR 99j:47013
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 R. Yang, The BergerShaw theorem in the Hardy module over the bidisk, J. Oper. Theory, 42 (1999), 379404. MR 2000h:47040
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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
DOI:
http://dx.doi.org/10.1090/S000299390206687X
PII:
S 00029939(02)06687X
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
