AMS :: St. Petersburg Mathematical Journal

Functions of perturbed pairs of noncommutative dissipative operators
HTML articles powered by AMS MathViewer

by A. B. Aleksandrov and V. V. Peller;
Translated by: the authors

St. Petersburg Math. J. 34 (2023), 379-392

DOI: https://doi.org/10.1090/spmj/1758

Published electronically: June 7, 2023

PDF | Request permission

Abstract:

Let a function $f$ belong to the inhomogeneous analytic Besov space $(Б_{\infty ,1}^{\,1})_+(\mathbb {R}^2)$. For a pair $(L,M)$ of not necessarily commuting maximal dissipative operators, the function $f(L,M)$ of $L$ and $M$ is defined as a densely defined linear operator. For $p\in [1,2]$, it is proved that if $(L_1,M_1)$ and $(L_2,M_2)$ are pairs of not necessarily commuting maximal dissipative operators such that both differences $L_1-L_2$ and $M_1-M_2$ belong to the Schatten–von Neumann class ${\boldsymbol S}_p$, then for an arbitrary function $f$ in $(Б_{\infty ,1}^{\,1})_+(\mathbb {R}^2)$, the operator difference $f(L_1,M_1)-f(L_2,M_2)$ belongs to ${\boldsymbol S}_p$ and the following Lipschitz type estimate holds: \begin{equation*} \|f(L_1,M_1)-f(L_2,M_2)\|_{{\boldsymbol S}_p} \le const\|f\|_{Б_{\infty ,1}^{\,1}}\max \big \{\|L_1-L_2\|_{{\boldsymbol S}_p},\|M_1-M_2\|_{{\boldsymbol S}_p}\big \}. \end{equation*}

References

A. B. Aleksandrov, F. L. Nazarov, and V. V. Peller, Functions of noncommuting self-adjoint operators under perturbation and estimates of triple operator integrals, Adv. Math. 295 (2016), 1–52. MR 3488031, DOI 10.1016/j.aim.2016.02.030
A. B. Aleksandrov and V. V. Peller, Functions of perturbed dissipative operators, Algebra i Analiz 23 (2011), no. 2, 9–51 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 23 (2012), no. 2, 209–238. MR 2841671, DOI 10.1090/S1061-0022-2012-01194-5
A. B. Aleksandrov and V. V. Peller, Operator Lipschitz functions, Uspekhi Mat. Nauk 71 (2016), no. 4(430), 3–106 (Russian, with Russian summary); English transl., Russian Math. Surveys 71 (2016), no. 4, 605–702. MR 3588921, DOI 10.4213/rm9729
A. B. Aleksandrov and V. V. Peller, Multiple operator integrals, Haagerup and Haagerup-like tensor products, and operator ideals, Bull. Lond. Math. Soc. 49 (2017), no. 3, 463–479. MR 3723631, DOI 10.1112/blms.12034
A. B. Aleksandrov and V. V. Peller, Dissipative operators and operator Lipschitz functions, Proc. Amer. Math. Soc. 147 (2019), no. 5, 2081–2093. MR 3937684, DOI 10.1090/proc/14335
A. B. Aleksandrov and V. V. Peller, Functions of perturbed commuting dissipative operators, Math. Nachr. 295 (2022), no. 6, 1042–1062. MR 4453949, DOI 10.1002/mana.202000239
A. B. Aleksandrov and V. V. Peller, Functions of perturbed commuting dissipative operators, Math. Nachr. 295 (2022), no. 6, 1042–1062. MR 4453949, DOI 10.1002/mana.202000239
Spectral theory and wave processes, Topics in Mathematical Physics, Vol. 1, Consultants Bureau Plenum Publishing Corporation, New York, 1967. Edited by M. Sh. Birman; Translated from the Russian. MR 210536
Spectral theory and problems in diffraction, Topics in Mathematical Physics, Vol. 2, Consultants Bureau, New York, 1968. Edited by M. Sh. Birman; Translated from Russian. MR 266727
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 348494
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 84745
K. Juschenko, I. G. Todorov, and L. Turowska, Multidimensional operator multipliers, Trans. Amer. Math. Soc. 361 (2009), no. 9, 4683–4720. MR 2506424, DOI 10.1090/S0002-9947-09-04771-0
M. Neumark, Spectral functions of a symmetric operator, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 4 (1940), 277–318 (Russian, with English summary). MR 2714
Jaak Peetre, New thoughts on Besov spaces, Duke University Mathematics Series, No. 1, Duke University, Mathematics Department, Durham, NC, 1976. MR 461123
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
V. V. Peller, Multiple operator integrals in perturbation theory, Bull. Math. Sci. 6 (2016), no. 1, 15–88. MR 3472849, DOI 10.1007/s13373-015-0073-y