Functions of commuting contractions under perturbation

Peller, Vladimir

doi:10.1002/mana.201800232

Cited by 9 publications

(6 citation statements)

References 14 publications

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“…We extend earlier results obtained for functions of self-adjoint operators and unitary operators (see [25] and [27]), functions of contractions (see [26] and [28]), functions of dissipative operators (see [6] and [8]), functions of normal operators (see [10]) and functions of commuting contractions (see [30]).…”

Section: Introductionsupporting

confidence: 87%

Functions of perturbed commuting dissipative operators

Aleksei

Peller

2022

Mathematische Nachrichten

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The main objective of the paper is to obtain sharp Lipschitz type estimates for the norm of operator differences for pairs and of commuting maximal dissipative operators. To obtain such estimates, we use double operator integrals with respect to semi‐spectral measures associated with the pairs and . Note that the situation is considerably more complicated than in the case of functions of two commuting contractions and to overcome difficulties we had to elaborate new techniques. We deduce from the main result Hölder type estimates for operator differences as well as their estimates in Schatten–von Neumann norms.

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Section: Introductionsupporting

confidence: 87%

Functions of perturbed commuting dissipative operators

Aleksei

Peller

2022

Mathematische Nachrichten

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“…We also obtain Hölder type estimates and estimates in Schatten-von Neumann norms S p . We extend earlier results obtained for functions of self-adjoint operators and unitary operators (see [Pe1] and [Pe3]), functions of contractions (see [Pe3] and [Pe4]), functions of dissipative operators (see [AP3] and [AP5]), functions of normal operators (see [APPS]) and functions of commuting contractions (see [Pe6]).…”

Section: Introductionsupporting

confidence: 86%

Functions of perturbed commuting dissipative operators

Aleksei¹,

Peller²

2020

Preprint

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The main objective of the paper is to obtain sharp Lipschitz type estimates for the norm of operator differences f (L1, M1) − f (L2, M2) for pairs (L1, M1) and (L2, M2) of commuting maximal dissipative operators. To obtain such estimates, we use double operator integrals with respect to semi-spectral measures associated with the pairs (L1, M1) and (L2, M2). Note that the situation is considerably more complicated than in the case of functions of two commuting contractions and to overcome difficulties we had to elaborate new techniques. We deduce from the main result Hölder type estimates for operator differences as well as their estimates in Schatten-von Neumann norms. Contents19 9. The principal inequality 19 10. Hölder type estimates and estimates in Schatten-von Neumann norms 22 References 23

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“…It follows from Lemma 3.2 of [Pe8] that under the above assumptions, inequalities (3.10) hold for Z j = α j (T 1 ), j ≥ 0, with M = sup ζ∈T j≥0 |α j (ζ)| 2 1/2 and for…”

Section: Triple Operator Integrals Haagerup Tensor Productsmentioning

confidence: 99%

Functions of perturbed pairs of noncommuting contractions

Aleksandrov¹,

Peller²

2020

Izv. Math.

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We consider functions f (T, R) of pairs of noncommuting contractions on Hilbert space and study the problem for which functions f we have Lipschitz type estimates in Schatten-von Neumann norms. We prove that if f belongs to the Besov class B 1 ∞,1 + (T 2 ) of nalytic functions in the bidisk, then we have a Lipschitz type estimate for functions f (T, R) of pairs of not necessarily commuting contractions (T, R) in the Schatten-von Neumann norms Sp for p ∈ [1, 2]. On the other hand, we show that for functions in B 1 ∞,1 + (T 2 ) there are no Lipschitz such type estimates for p > 2 as well as in the operator norm. ContentsThe research of the first author is supposed by RFBR grant 17-01-00607. The publication was prepared with the support of the RUDN University Program 5-100.Corresponding author: V.V. Peller; email: peller@math.msu.edu. 1 This paper can be considered as a continuation of the results of [Pe1]-[Pe7], [AP1]-[AP4], [AP6], [APPS], [NP], [ANP], [PS] and [KPSS] for functions of perturbed selfadjoint operators, contractions, normal operators, dissipative operators, functions of collections of commuting operators and functions of collections of noncommuting operators.Recall that a Lipschitz function f on R does not have to be operator Lipschitz, i.e., the condition |f (x) − f (y)| ≤ const |x − y|, x, y ∈ R, does not imply thatfor arbitrary self-adjoint operators (bounded or unbounded, does not matter) A and B. This was first established in [F].It turned out that functions in the (homogeneous) Besov space B 1 ∞,1 (R) are operator Lipschitz; this was established in [Pe1] and [Pe3] (see [Pee] for detailed information about Besov classes). We refer the reader to the recent survey [AP4] for detailed information on operator Lipschitz functions. In particular, [AP4] presents various sufficient conditions and necessary conditions for a function on R to be operator Lipschitz. It is well known that if f is an operator Lipschitz function on R, and A and B are self-adjoint operators such that the difference A − B belongs to the Schatten-von Neumann class S p ,

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Functions of commuting contractions under perturbation

Cited by 9 publications

References 14 publications

Functions of perturbed commuting dissipative operators

Functions of perturbed commuting dissipative operators

Functions of perturbed commuting dissipative operators

Functions of perturbed pairs of noncommuting contractions

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