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Mat. Zametki, 2019, Volume 106, Issue 4, Pages 483–490 (Mi mz12259)  

On a Trace Formula for Functions of Noncommuting Operators

A. B. Aleksandrova, V. V. Pellerbc, D. S. Potapovd

a St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
b Michigan State University, Department of Mathematics
c Peoples' Friendship University of Russia, Moscow
d University of New South Wales

Abstract: The main result of the paper is that the Lifshits–Krein trace formula cannot be generalized to the case of functions of noncommuting self-adjoint operators. To prove this, we show that, for pairs $(A_1,B_1)$ and $(A_2,B_2)$ of bounded self-adjoint operators with trace class differences $A_2-A_1$ and $B_2-B_1$, it is impossible to estimate the modulus of the trace of the difference $f(A_2,B_2)-f(A_1,B_1)$ in terms of the norm of $f$ in the Lipschitz class.

Keywords: trace, trace class operators, operators Lipschitz functions, Lifshits–Krein trace formula.

Funding Agency Grant Number
Russian Foundation for Basic Research 17-01-00607
Australian Research Council
Ministry of Education and Science of the Russian Federation 5-100
The research of the first author was supported in part by the Russian Foundation for Basic Research under grant 17-01-00607. The publication was prepared with the support of the “RUDN University Program 5-100.” The research of the third author was supported in part by ARC.

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/mzm12259

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English version:
Mathematical Notes, 2019, 106:4, 481–487

Bibliographic databases:

UDC: 517
Received: 24.11.2018

Citation: A. B. Aleksandrov, V. V. Peller, D. S. Potapov, “On a Trace Formula for Functions of Noncommuting Operators”, Mat. Zametki, 106:4 (2019), 483–490; Math. Notes, 106:4 (2019), 481–487

Citation in format AMSBIB
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\paper On a Trace Formula for Functions of Noncommuting Operators
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\pages 483--490
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\crossref{https://doi.org/10.4213/mzm12259}
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\pages 481--487
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