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Published 2008 | Submitted
Journal Article Open

On the Koplienko Spectral Shift Function. I. Basics

Abstract

We study the Koplienko Spectral Shift Function (KoSSF), which is distinct from the one of Krein (KrSSF). KoSSF is defined for pairs A,B with (A − B) ∈ I2, the Hilbert–Schmidt operators, while KrSSF is defined for pairs A,B with (A−B) ∈ I1, the trace class operators. We review various aspects of the construction of both KoSSF and KrSSF. Among our new results are: (i) that any positive Riemann integrable function of compact support occurs as a KoSSF; (ii) that there exist A,B with (A−B) ∈ I2 so det2((A − z)(B − z)^−1) does not have nontangential boundary values; (iii) an alternative definition of KoSSF in the unitary case; and (iv) a new proof of the invariance of the a.c. spectrum under I1-perturbations that uses the KrSSF.

Additional Information

Published version. Copyright ILTPE. 2008. Preprint copyright 2007, The Authors. Submitted [to arXiv] 24 May 2007. Submitted to the Marchenko and Pastur birthday issue of Journal of Mathematical Physics, Analysis and Geometry. We are indebted to E. Lieb, K.A. Makarov, V.V. Peller, and M.B. Ruskai for useful discussions. F. G. and A.P. wish to thank Gary Lorden and Tom Tombrello for the hospitality of Caltech where some of this work was done. F.G. gratefully acknowledges a research leave for the academic year 2005/06 granted by the Research Council and the Office of Research of the University of Missouri-Columbia. A.P. gratefully acknowledges financial support by the Leverhulme Trust. It is a great pleasure to dedicate this paper to the birthdays of two giants of spectral theory: Vladimir A. Marchenko and Leonid A. Pastur. [F.G. was] [s]upported in part by NSF Grant DMS-0405526. [A.P. was] [s]upported in part by the Leverhulme Trust. [B.S. was] [s]upported in part by NSF Grant DMS-0140592 and U.S.–Israel Binational Science Foundation (BSF) Grant No. 2002068.

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