The Lifshitz-Krein trace formula and operator Lipschitz functions

Author:
V. V. Peller

Journal:
Proc. Amer. Math. Soc. **144** (2016), 5207-5215

MSC (2010):
Primary 47A55, 47B10; Secondary 47B15, 47B25, 47A60, 47B49

DOI:
https://doi.org/10.1090/proc/13140

Published electronically:
August 1, 2016

MathSciNet review:
3556265

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We solve a problem by M.G. Krein and describe the maximal class of functions on the real line, for which the Lifshitz-Krein trace formula holds for arbitrary self-adjoint operators and with in the trace class . We prove that this class of functions coincideS with the class of operator Lipschitz functions.

**[AP1]**A. B. Aleksandrov and V. V. Peller,*Operator Hölder-Zygmund functions*, Adv. Math.**224**(2010), no. 3, 910–966. MR**2628799**, https://doi.org/10.1016/j.aim.2009.12.018- [AP2]
A. B. Aleksandrov and V. V. Peller,
*Operator Lipschitz functions*, to appear in Russian Math. Surveys. **[BS1]**M. Š. Birman and M. Z. Solomjak,*Double Stieltjes operator integrals and problems on multipliers*, Dokl. Akad. Nauk SSSR**171**(1966), 1251–1254 (Russian). MR**0209871****[BS2]**M. Š. Birman and M. Z. Solomjak,*Double Stieltjes operator integrals. II*, Problems of Mathematical Physics, No. 2, Spectral Theory, Diffraction Problems (Russian), Izdat. Leningrad. Univ., Leningrad, 1967, pp. 26–60 (Russian). MR**0234304****[BS3]**M. Š. Birman and M. Z. Solomjak,*Remarks on the spectral shift function*, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)**27**(1972), 33–46 (Russian). Boundary value problems of mathematical physics and related questions in the theory of functions, 6. MR**0315482****[BS4]**M. Š. Birman and M. Z. Solomjak,*Double Stieltjes operator integrals. III*, Problems of mathematical physics, No. 6 (Russian), Izdat. Leningrad. Univ., Leningrad, 1973, pp. 27–53 (Russian). MR**0348494****[BS5]**M. Birman and M. Solomyak,*Tensor product of a finite number of spectral measures is always a spectral measure*, Integral Equations Operator Theory**24**(1996), no. 2, 179–187. MR**1371945**, https://doi.org/10.1007/BF01193459**[DK]**Yu. L. Daleckiĭ and S. G. Kreĭn,*Integration and differentiation of functions of Hermitian operators and applications to the theory of perturbations*, Voronež. Gos. Univ. Trudy Sem. Funkcional. Anal.**1956**(1956), no. 1, 81–105 (Russian). MR**0084745****[F]**Ju. B. Farforovskaja,*An example of a Lipschitzian function of selfadjoint operators that yields a nonnuclear increase under a nuclear perturbation*, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)**30**(1972), 146–153 (Russian). Investigations of linear operators and the theory of functions, III. MR**0336400****[JW]**B. E. Johnson and J. P. Williams,*The range of a normal derivation*, Pacific J. Math.**58**(1975), no. 1, 105–122. MR**380490****[Ka]**Tosio Kato,*Continuity of the map 𝑆\mapsto\mid𝑆\mid for linear operators*, Proc. Japan Acad.**49**(1973), 157–160. MR**405148****[KS]**Edward Kissin and Victor S. Shulman,*Classes of operator-smooth functions. I. Operator-Lipschitz functions*, Proc. Edinb. Math. Soc. (2)**48**(2005), no. 1, 151–173. MR**2117717**, https://doi.org/10.1017/S0013091503000178**[KPSS]**E. Kissin, D. Potapov, V. Shulman, and F. Sukochev,*Operator smoothness in Schatten norms for functions of several variables: Lipschitz conditions, differentiability and unbounded derivations*, Proc. Lond. Math. Soc. (3)**105**(2012), no. 4, 661–702. MR**2989800**, https://doi.org/10.1112/plms/pds014**[Kr]**M. G. Kreĭn,*On the trace formula in perturbation theory*, Mat. Sbornik N.S.**33(75)**(1953), 597–626 (Russian). MR**0060742****[L]**I. M. Lifšic,*On a problem of the theory of perturbations connected with quantum statistics*, Uspehi Matem. Nauk (N.S.)**7**(1952), no. 1(47), 171–180 (Russian). MR**0049490****[Mc]**Alan McIntosh,*Counterexample to a question on commutators*, Proc. Amer. Math. Soc.**29**(1971), 337–340. MR**276798**, https://doi.org/10.1090/S0002-9939-1971-0276798-4**[Pee]**Jaak Peetre,*New thoughts on Besov spaces*, Mathematics Department, Duke University, Durham, N.C., 1976. Duke University Mathematics Series, No. 1. MR**0461123****[Pe1]**V. V. Peller,*Hankel operators of class 𝔖_{𝔭} and their applications (rational approximation, Gaussian processes, the problem of majorization of operators)*, Mat. Sb. (N.S.)**113(155)**(1980), no. 4(12), 538–581, 637 (Russian). MR**602274****[Pe2]**V. V. Peller,*Hankel operators in the theory of perturbations of unitary and selfadjoint operators*, Funktsional. Anal. i Prilozhen.**19**(1985), no. 2, 37–51, 96 (Russian). MR**800919****[Pe3]**Vladimir V. Peller,*Hankel operators in the perturbation theory of unbounded selfadjoint operators*, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 529–544. MR**1044807****[Pe4]**Vladimir V. Peller,*Hankel operators and their applications*, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. MR**1949210**- [Pe5]
V. V. Peller,
*Multiple operator integrals in perturbation theory*, Bull. Math. Sci.**6**(2016), 15-88. **[Pi]**Gilles Pisier,*Similarity problems and completely bounded maps*, Second, expanded edition, Lecture Notes in Mathematics, vol. 1618, Springer-Verlag, Berlin, 2001. Includes the solution to “The Halmos problem”. MR**1818047**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
47A55,
47B10,
47B15,
47B25,
47A60,
47B49

Retrieve articles in all journals with MSC (2010): 47A55, 47B10, 47B15, 47B25, 47A60, 47B49

Additional Information

**V. V. Peller**

Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

DOI:
https://doi.org/10.1090/proc/13140

Received by editor(s):
January 7, 2016

Received by editor(s) in revised form:
February 5, 2016

Published electronically:
August 1, 2016

Additional Notes:
The author was partially supported by NSF grant DMS 1300924

Communicated by:
Pamela B. Gorkin

Article copyright:
© Copyright 2016
American Mathematical Society