Pairs of selfadjoint operators and their invariants

Alpay, D.; Gohberg, I.

doi:10.1090/S1061-0022-04-00844-1

Pairs of selfadjoint operators and their invariants
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by D. Alpay and I. Gohberg

St. Petersburg Math. J. 16 (2005), 59-104

DOI: https://doi.org/10.1090/S1061-0022-04-00844-1

Published electronically: December 14, 2004

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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 Kreĭn’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 Kreĭn’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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