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 Mat. Sb., 1994, Volume 185, Number 10, Pages 91–144 (Mi msb934)

The spectral shift function, the characteristic function of a contraction, and a generalized integral

A. V. Rybkin

Abstract: Let $T$ be a contraction that is a trace class perturbation of a unitary operator $V$, and let $\{\lambda_k\}$ be the discrete spectrum of $T$. For a sufficiently large class of functions $\Phi$ the trace formula
$$\operatorname{tr}\{\Phi(T)-\Phi (V)\}=\sum_k\{\Phi(\lambda_k)-\Phi(\lambda_k/|\lambda_k|)\}+(B)\int_0^{2\pi}\Phi'(e^{i\varphi}) d\Omega(\varphi),$$
holds. This formula is a direct analogue of the well-known M. G. Krein trace formula for unitary operators. It is natural to call the function $\Omega$ the spectral shift distribution. Generally speaking, it is not of bounded variation; however, the integral in the trace formula exists in the wider $B$-sense. In the present paper an explicit representation is obtained for $\Omega$ in terms of the characteristic function $\Theta(\lambda)$ of the contraction $T$, and also a relation between a certain derivative $\Omega'$ and the scattering matrix $S(\varphi)$ of the pair $(T,V)$:
$$\det S(\varphi)=\exp\{-2\pi i\overline{\Omega'(\varphi)} \} \quad \textrm{a.e. with respect to Lebesgue measure}$$
is established. A necessary and sufficient condition that $\Omega$ have bounded variation is obtained. In particular, the necessary and sufficient condition requires that the singular spectrum of the contraction $T$ be empty. The main results are complete.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1995, 83:1, 237–281

Bibliographic databases:

UDC: 517
MSC: Primary 47A45, 47A60; Secondary 47A40

Citation: A. V. Rybkin, “The spectral shift function, the characteristic function of a contraction, and a generalized integral”, Mat. Sb., 185:10 (1994), 91–144; Russian Acad. Sci. Sb. Math., 83:1 (1995), 237–281

Citation in format AMSBIB
\Bibitem{Ryb94} \by A.~V.~Rybkin \paper The spectral shift function, the~characteristic function of a~contraction, and a~generalized integral \jour Mat. Sb. \yr 1994 \vol 185 \issue 10 \pages 91--144 \mathnet{http://mi.mathnet.ru/msb934} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1309184} \zmath{https://zbmath.org/?q=an:0852.47004} \transl \jour Russian Acad. Sci. Sb. Math. \yr 1995 \vol 83 \issue 1 \pages 237--281 \crossref{https://doi.org/10.1070/SM1995v083n01ABEH003589} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995TQ10000012} 

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This publication is cited in the following articles:
1. Alexei Rybkin, “On a trace formula of the Buslaev–Faddeev type for a long-range potential”, J Math Phys (N Y ), 40:3 (1999), 1334
2. S. A. M. Marcantognini, M. D. Morán, “Koplienko–Neidhardt trace formula for pairs of contraction operators and pairs of maximal dissipative operators”, Math Nachr, 279:7 (2006), 784
3. V. A. Sadovnichii, V. E. Podolskii, “Traces of operators”, Russian Math. Surveys, 61:5 (2006), 885–953
4. F. Gesztesy, A. Pushnitski, B. Simon, “On the Koplienko spectral shift function. I. Basics”, Zhurn. matem. fiz., anal., geom., 4:1 (2008), 63–107
5. Potapov D. Sukochev F., “Koplienko Spectral Shift Function on the Unit Circle”, Commun. Math. Phys., 309:3 (2012), 693–702
6. M. M. Malamud, H. Neidhardt, V. V. Peller, “Analytic operator Lipschitz functions in the disk and a trace formula for functions of contractions”, Funct. Anal. Appl., 51:3 (2017), 185–203
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