Analytic operator Lipschitz functions in the disk and a trace formula for functions of contractions

M. M. Malamud ; H. Neidhardt ; V. V. Peller

Funkcionalʹnyj analiz i ego priloženiâ, Tome 51 (2017) no. 3, pp. 33-55.

Résumé

In this paper we prove that for an arbitrary pair $\{T_1,T_0\}$ of contractions on Hilbert space with trace class difference, there exists a function $\boldsymbol\xi$ in $L^1(\mathbb{T})$ (called a spectral shift function for the pair $\{T_1,T_0\}$) such that the trace formula $\operatorname{trace}(f(T_1)-f(T_0))=\int_{\mathbb{T}} f'(\zeta)\boldsymbol{\xi}(\zeta)\,d\zeta$ holds for an arbitrary operator Lipschitz function $f$ analytic in the unit disk.

Détail

Keywords: contraction, dissipative operator, spectral shift function, operator Lipschitz functions
Mots-clés : trace formulae, perturbation determinant.

Bibliographie

[1] V. M. Adamjan, H. Neidhardt, “On the summability of the spectral shift function for pair of contractions and dissipative operators”, J. Operator Theory, 24:1 (1990), 187–205 | MR

[2] A. B. Aleksandrov, V. V. Peller, “Operatorno lipshitsevy funktsii”, UMN, 71:4 (2016), 3–106 | DOI | MR | Zbl

[3] A. B. Aleksandrov, V. V. Peller, “Formula sledov M.Kreina dlya unitarnykh operatorov i operatorno lipshitsevy funktsii”, Funkts. analiz i ego pril., 50:3 (2016), 1–11 | DOI | MR | Zbl

[4] M. Sh. Birman, M. Z. Solomyak, “Dvoinye operatornye integraly Stiltesa”, Problemy matematicheskoi fiziki, v. 1, Spektralnaya teoriya i volnovye protsescy, Izd-vo LGU, 1966, 33–67 | MR

[5] M. Sh. Birman, M. Z. Solomyak, “Zamechaniya o funktsii spektralnogo sdviga”, Zap. nauchn. sem. LOMI, 27 (1972), 33–46 | Zbl

[6] Yu. L. Daletskii, S. G. Krein, “Integrirovanie i differentsirovanie ermitovykh operatorov i prilozhenie k teorii vozmuschenii”, Trudy seminara po funkts. analizu, 1, Voronezhskii gos. un-t, 1956, 81–105

[7] V. A. Derkach, M. M. Malamud, “Generalized resolvents and the boundary value problems for Hermitian operators with gaps”, J. Funct. Anal., 95 (1991), 1–95 | DOI | MR | Zbl

[8] Yu. B. Farforovskaya, “Primer lipshitsevoi funktsii ot samosopryazhennykh operatorov”, Zap. nauchn. sem. LOMI, 30 (1972), 146–153 | MR | Zbl

[9] F. Gesztezy, K. A. Makarov, S. N. Naboko, “The spectral shift operator”, Math. Results in Quantum Mechanic (Prague, 1998), Oper. Theory Adv. Appl., 108, Birkhäuser, Basel, 1999, 59–90 | MR | Zbl

[10] B. E. Johnson, J. P. Williams, “The range of a normal derivation”, Pacific J. Math., 58:1 (1975), 105–122 | DOI | MR | Zbl

[11] E. Kissin, V. S. Shulman, “Classes of operator-smooth functions. I. Operator-Lipschitz functions”, Proc. Edinb. Math. Soc., 48:1 (2005), 151–173 | DOI | MR | Zbl

[12] P. Koosis, Introduction to $H\sb{p}$ spaces, London Mathematical Society Lecture Note Series, 40, Cambridge University Press, Cambridge, 1980 | MR

[13] E. Korotyaev, A. Laptev, Trace formulae for Schrödinger operators with complex-valued potentials on cubic lattices, arXiv: 1609.09703 | MR

[14] M. G. Krein, “O formule sledov v teorii vozmuschenii”, Matem. sb., 33:3 (1953), 597–626 | Zbl

[15] M. G. Krein, “Ob opredelitelyakh vozmuscheniya i formule sledov dlya unitarnykh i samosopryazhennykh operatorov”, Dokl. AN SSSR, 144:2 (1962), 268–271 | Zbl

[16] M. G. Krein, “O nekotorykh novykh issledovaniyakh po teorii vozmuschenii samosopryazhennykh operatorov”, Pervaya letnyaya matem. shkola, v. I, Naukova Dumka, Kiev, 1964, 103–187

[17] M. G. Krein, “Ob opredelitelyakh vozmuscheniya i formule sledov dlya nekotorykh klassov par operatorov”, J. Operator Theory, 17:1 (1987), 129–187 | MR | Zbl

[18] I. M. Lifshits, “Ob odnoi zadache teorii vozmuschenii, svyazannoi s kvantovoi statistikoi”, UMN, 7:1(47) (1952), 171–180 | MR | Zbl

[19] M. Malamud, H. Neidhardt, “Perturbation determinants for singular perturbations”, Russian J. Math. Phys., 21:1 (2014), 55–98 | DOI | MR | Zbl

[20] M. Malamud, H. Neidhardt, “Trace formulas for additive and non-additive perturbations”, Adv. Math., 274 (2015), 736–832 ; arXiv: 1212.6887 | DOI | MR | Zbl

[21] K. Makarov, A. Skripka, M. Zinchenko, “On a perturbation determinant for accumulative operators”, Integral Equations Operator Theory, 81:1 (2015), 301–317 | DOI | MR | Zbl

[22] V. V. Peller, “Operatory Gankelya v teorii vozmuschenii unitarnykh i samosopryazhennykh operatorov”, Funkts. analiz i ego pril., 19:2 (1985), 37–51 | MR | Zbl

[23] V. V. Peller, “For which $f$ does $A-B\inS_{p}$ imply that $f(A)-f(B)\inS_{p}$?”, Operators in indefinite metric spaces, scattering theory and other topics (10th Int. Conf., Bucharest/Rom., 1985), Operator Theory Adv. Appl., 24, 1987, 289–294 | MR | Zbl

[24] V. V. Peller, “Hankel operators in the perturbation theory of unbounded self-adjoint operators”, Analysis and Partial Differential Equations, Lecture Notes in Pure and Appl. Math., 122, Dekker, New York, 1990, 529–544 | MR

[25] V. V. Peller, “Differentiability of functions of contractions”, Linear and complex analysis, Amer. Math. Soc. Translations, Ser. 2, 226, Amer. Math. Soc., Providence, RI, 2009, 109–131 | DOI | MR | Zbl

[26] V. V. Peller, “The Lifshits–Krein trace formula and operator Lipschitz functions”, Proc. Amer. Math. Soc., 144:12 (2016), 5207–5215 | DOI | MR | Zbl

[27] A. V. Rybkin, “Funktsiya spektralnogo sdviga dlya dissipativnogo i samosopryazhennogo operatorov i formula sledov dlya rezonansov”, Matem. sb., 125(167):3 (1984), 420–430 | MR | Zbl

[28] A. V. Rybkin, “Formula sledov dlya szhimayuschego i unitarnogo operatorov”, Funkts. analiz i ego pril., 21:4 (1987), 85–87 | MR

[29] A. V. Rybkin, “Diskretnyi i singulyarnyi spektr v formule sledov dlya szhimayuschego i unitarnogo operatorov”, Funkts. analiz i ego pril., 23:3 (1989), 84–85 | MR

[30] A. V. Rybkin, “Funktsiya spektralnogo sdviga, kharakteristicheskaya funktsiya szhatiya i obobschennyi integral”, Matem. sb., 185:10 (1994), 91–144 | Zbl

[31] B. Sëkefalvi-Nad, Ch. Foyash, Garmonicheskii analiz operatorov v gilbertovom prostranstve, Mir, M., 1970 | MR

[32] D. R. Yafaev, Mathematical scattering theory, Translations of Mathematical Monographs, 105, Amer. Math. Soc., Providence, RI, 1992 | DOI | MR | Zbl

[33] K. Iosida, Funktsionalnyi analiz, Mir, M., 1967 | MR