Trace formulae for perturbations of class $\boldsymbol{{\boldsymbol S}_m}$

Aleksandrov, A. B.; Peller, Vladimir

doi:10.4171/jst/1

Cited by 14 publications

(37 citation statements)

References 11 publications

(38 reference statements)

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“…Note that the proof of Theorem 2.4.1 given in [10] contains an inaccuracy. We give here a corrected proof.…”

Section: In This Formula By T (M)mentioning

confidence: 99%

“…First, formula (2.4.2) was extended for arbitrary functions f in the Besov class B m ∞,1 (R). Secondly, much more general trace formulae for perturbations of class S m we obtained in [10].…”

Section: Trace Formulae For Perturbations Of Class Smentioning

confidence: 99%

“…Note that Theorem 2.4.1 was used in [10] to obtain considerably more general trace formulae. In particular trace formulae were found for…”

Section: In This Formula By T (M)mentioning

confidence: 99%

“…for functions f on R such that the derivatives f The results of [62] were improved in [10]. First, formula (2.4.2) was extended for arbitrary functions f in the Besov class B m ∞,1 (R).…”

Section: Trace Formulae For Perturbations Of Class Smentioning

confidence: 99%

“…It was shown in [10] that the Taylor approximation admits the following representation in terms of the multiple operator integral:…”

Section: Trace Formulae For Perturbations Of Class Smentioning

confidence: 99%

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Multiple operator integrals in perturbation theory

Peller

2015

Bull. Math. Sci.

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The purpose of this survey article is to give an introduction to double operator integrals and multiple operator integrals and to discuss various applications of such operator integrals in perturbation theory. We start with the Birman-Solomyak approach to define double operator integrals and consider applications in estimating operator differences f (A)− f (B) for self-adjoint operators A and B. Next, we present the Birman-Solomyak approach to the Lifshits-Krein trace formula that is based on double operator integrals. We study the class of operator Lipschitz functions, operator differentiable functions, operator Hölder functions, obtain Schatten-von Neumann estimates for operator differences. Finally, we consider in Chapter 1 estimates of functions of normal operators and functions of d-tuples of commuting self-adjoint operators under perturbations. In Chapter 2 we define multiple operator integrals in the case when the integrands belong to the integral projective tensor product of L ∞ spaces. We consider applications of such multiple operator integrals to the problem of the existence of higher operator derivatives and to the problem of estimating higher operator differences. We also consider connections with trace formulae for functions of operators under perturbations of class S m , m ≥ 2. In the last chapter we define Haagerup-like tensor products of the first kind and of the second kind and we use them to study functions of noncommuting self-adjoint operators under perturbation. We show that for functions f in the Besov class B

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“…Note that the proof of Theorem 2.4.1 given in [10] contains an inaccuracy. We give here a corrected proof.…”

Section: In This Formula By T (M)mentioning

confidence: 99%

Section: Trace Formulae For Perturbations Of Class Smentioning

confidence: 99%

“…Note that Theorem 2.4.1 was used in [10] to obtain considerably more general trace formulae. In particular trace formulae were found for…”

Section: In This Formula By T (M)mentioning

confidence: 99%

“…for functions f on R such that the derivatives f The results of [62] were improved in [10]. First, formula (2.4.2) was extended for arbitrary functions f in the Besov class B m ∞,1 (R).…”

Section: Trace Formulae For Perturbations Of Class Smentioning

confidence: 99%

“…It was shown in [10] that the Taylor approximation admits the following representation in terms of the multiple operator integral:…”

Section: Trace Formulae For Perturbations Of Class Smentioning

confidence: 99%

See 3 more Smart Citations

Multiple operator integrals in perturbation theory

Peller

2015

Bull. Math. Sci.

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Second‐order trace formulas

Chattopadhyay,

Das,

Pradhan

2024

Mathematische Nachrichten

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Koplienko [Sib. Mat. Zh. 25 (1984), 62–71; English transl. in Siberian Math. J. 25 (1984), 735–743] found a trace formula for perturbations of self‐adjoint operators by operators of Hilbert–Schmidt class . Later, Neidhardt introduced a similar formula in the case of pairs of unitaries via multiplicative path in [Math. Nachr. 138 (1988), 7–25]. In 2012, Potapov and Sukochev [Comm. Math. Phys. 309 (2012), no. 3, 693–702] obtained a trace formula like the Koplienko trace formula for pairs of contractions by answering an open question posed by Gesztesy, Pushnitski, and Simon [Zh. Mat. Fiz. Anal. Geom. 4 (2008), no. 1, 63–107, 202; Open Question 11.2]. In this paper, we supply a new proof of the Koplienko trace formula in the case of pairs of contractions , where the initial operator is normal, via linear path by reducing the problem to a finite‐dimensional one as in the proof of Krein's trace formula by Voiculescu [Oper. Theory Adv. Appl. 24 (1987) 329–332] and Sinha and Mohapatra [Proc. Indian Acad. Sci. Math. Sci. 104 (1994), no. 4, 819–853] and [Integral Equations Operator Theory 24 (1996), no. 3, 285–297]. Consequently, we obtain the Koplienko trace formula for a class of pairs of contractions using the Schäffer matrix unitary dilation. Moreover, we also obtain the Koplienko trace formula for a pair of self‐adjoint operators and maximal dissipative operators using the Cayley transform. At the end, we extend the Koplienko–Neidhardt trace formula for a class of pairs of contractions via multiplicative path using the finite‐dimensional approximation method.

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Applications

Skripka¹,

Tomskova²

2019

Multilinear Operator Integrals

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Trace formulae for perturbations of class $\boldsymbol{{\boldsymbol S}_m}$

Cited by 14 publications

References 11 publications

Multiple operator integrals in perturbation theory

Multiple operator integrals in perturbation theory

Second‐order trace formulas

Applications

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