The Lifshitz–Krein trace formula and operator Lipschitz functions

Peller, V.

doi:10.1090/proc/13140

The Lifshitz–Krein trace formula and operator Lipschitz functions
HTML articles powered by AMS MathViewer

by V. V. Peller PDF

Proc. Amer. Math. Soc. 144 (2016), 5207-5215 Request permission

Abstract:

We solve a problem by M.G. Krein and describe the maximal class of functions $f$ on the real line, for which the Lifshitz–Krein trace formula $\mathrm {trace}(f(A)-f(B))=\int _{\mathbb {R}} f’(s)\boldsymbol {\xi }(s) ds$ holds for arbitrary self-adjoint operators $A$ and $B$ with $A-B$ in the trace class $\boldsymbol {S}_1$. We prove that this class of functions coincideS with the class of operator Lipschitz functions.

References

A. B. Aleksandrov and V. V. Peller, Operator Hölder-Zygmund functions, Adv. Math. 224 (2010), no. 3, 910–966. MR 2628799, DOI 10.1016/j.aim.2009.12.018
A. B. Aleksandrov and V. V. Peller, Operator Lipschitz functions, to appear in Russian Math. Surveys.
M. Š. Birman and M. Z. Solomjak, Double Stieltjes operator integrals, Probl. Math. Phys., No. I, Spectral Theory and Wave Processes (Russian), Izdat. Leningrad. Univ., Leningrad, 1966, pp. 33–67 (Russian). MR 0209872
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
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
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
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, DOI 10.1007/BF01193459
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
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
B. E. Johnson and J. P. Williams, The range of a normal derivation, Pacific J. Math. 58 (1975), no. 1, 105–122. MR 380490
Tosio Kato, Continuity of the map $S\mapsto \mid S\mid$ for linear operators, Proc. Japan Acad. 49 (1973), 157–160. MR 405148
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, DOI 10.1017/S0013091503000178
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, DOI 10.1112/plms/pds014
M. G. Kreĭn, On the trace formula in perturbation theory, Mat. Sbornik N.S. 33(75) (1953), 597–626 (Russian). MR 0060742
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
Alan McIntosh, Counterexample to a question on commutators, Proc. Amer. Math. Soc. 29 (1971), 337–340. MR 276798, DOI 10.1090/S0002-9939-1971-0276798-4
Jaak Peetre, New thoughts on Besov spaces, Duke University Mathematics Series, No. 1, Duke University, Mathematics Department, Durham, N.C., 1976. MR 0461123
V. V. Peller, Hankel operators of class ${\mathfrak {S}}_{p}$ 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
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
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
Vladimir V. Peller, Hankel operators and their applications, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. MR 1949210, DOI 10.1007/978-0-387-21681-2
V. V. Peller, Multiple operator integrals in perturbation theory, Bull. Math. Sci. 6 (2016), 15-88.
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, DOI 10.1007/b55674