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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Pairs of selfadjoint operators and their invariants

Authors: D. Alpay and I. Gohberg
Original publication: Algebra i Analiz, tom 16 (2004), nomer 1.
Journal: St. Petersburg Math. J. 16 (2005), 59-104
MSC (2000): Primary 34L25, 81U40, 47A56
Published electronically: December 14, 2004
MathSciNet review: 2069002
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Abstract: A trace formula is proved for pairs of selfadjoint operators that are close to each other in a certain sense. An important role is played by a function analytic in the open upper half-plane and with positive imaginary part there. This function, called the characteristic function of the pair, coincides with Kre{\u{\i}}\kern.15emn's $Q$-function in the case where the selfadjoint operators are canonical extensions of a common simple and closed Hermitian operator. Special emphasis is given to the finite-dimensional case. Relationships with Kre{\u{\i}}\kern.15emn's spectral shift function are also considered. Finally, the case of canonical differential expressions is discussed briefly. In this case, the function ${N}$ may be chosen to be the Weyl function of the canonical differential expression.

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Additional Information

D. Alpay
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, POB 653, Beer-Sheva 84105, Israel

I. Gohberg
Affiliation: School of Mathematical Sciences, The Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv, Ramat-Aviv 69989, Israel

Keywords: Kre\u{\i}n's spectral shift function, the $Q$-function associated with a symmetric operator, the Weyl function
Received by editor(s): October 24, 2003
Published electronically: December 14, 2004
Additional Notes: The research of the second author was supported by the Israel Science Foundation (grant no. 322/00)
Dedicated: Dedicated to Mikhail Birman on the occasion of his 75th birthday, with admiration
Article copyright: © Copyright 2004 American Mathematical Society