Functions of noncommuting self-adjoint operators under perturbation and estimates of triple operator integrals

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Functions of noncommuting self-adjoint operators under perturbation and estimates of triple operator integrals. / Александров, Алексей Борисович; Nazarov, F. L.; Peller, V. V.

In: Advances in Mathematics, Vol. 295, 04.06.2016, p. 1-52.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{69cd0da2d02e45d88e6d7765467fb883,

title = "Functions of noncommuting self-adjoint operators under perturbation and estimates of triple operator integrals",

abstract = "We define functions of noncommuting self-adjoint operators with the help of double operator integrals. We are studying the problem to find conditions on a function f on R2, for which the map (A, B)↦f(A, B) is Lipschitz in the operator norm and in Schatten-von Neumann norms Sp. It turns out that for functions f in the Besov class B∞,11(R2), the above map is Lipschitz in the Sp norm for p∈[1, 2]. However, it is not Lipschitz in the operator norm, nor in the Sp norm for p>2. The main tool is triple operator integrals. To obtain the results, we introduce new Haagerup-like tensor products of L∞ spaces and obtain Schatten-von Neumann norm estimates of triple operator integrals. We also obtain similar results for functions of noncommuting unitary operators.",

keywords = "Functions of noncommuting operators, Haagerup tensor products, Haagerup-like tensor products, Lipschitz type estimates, Schatten-von Neumann classes, Triple operator integrals",

author = "Александров, {Алексей Борисович} and Nazarov, {F. L.} and Peller, {V. V.}",

note = "Publisher Copyright: {\textcopyright} 2016 Elsevier Inc.",

year = "2016",

month = jun,

day = "4",

doi = "10.1016/j.aim.2016.02.030",

language = "English",

volume = "295",

pages = "1--52",

journal = "Advances in Mathematics",

issn = "0001-8708",

publisher = "Elsevier",

}

RIS

TY - JOUR

T1 - Functions of noncommuting self-adjoint operators under perturbation and estimates of triple operator integrals

AU - Александров, Алексей Борисович

AU - Nazarov, F. L.

AU - Peller, V. V.

PY - 2016/6/4

Y1 - 2016/6/4

N2 - We define functions of noncommuting self-adjoint operators with the help of double operator integrals. We are studying the problem to find conditions on a function f on R2, for which the map (A, B)↦f(A, B) is Lipschitz in the operator norm and in Schatten-von Neumann norms Sp. It turns out that for functions f in the Besov class B∞,11(R2), the above map is Lipschitz in the Sp norm for p∈[1, 2]. However, it is not Lipschitz in the operator norm, nor in the Sp norm for p>2. The main tool is triple operator integrals. To obtain the results, we introduce new Haagerup-like tensor products of L∞ spaces and obtain Schatten-von Neumann norm estimates of triple operator integrals. We also obtain similar results for functions of noncommuting unitary operators.

AB - We define functions of noncommuting self-adjoint operators with the help of double operator integrals. We are studying the problem to find conditions on a function f on R2, for which the map (A, B)↦f(A, B) is Lipschitz in the operator norm and in Schatten-von Neumann norms Sp. It turns out that for functions f in the Besov class B∞,11(R2), the above map is Lipschitz in the Sp norm for p∈[1, 2]. However, it is not Lipschitz in the operator norm, nor in the Sp norm for p>2. The main tool is triple operator integrals. To obtain the results, we introduce new Haagerup-like tensor products of L∞ spaces and obtain Schatten-von Neumann norm estimates of triple operator integrals. We also obtain similar results for functions of noncommuting unitary operators.

KW - Functions of noncommuting operators

KW - Haagerup tensor products

KW - Haagerup-like tensor products

KW - Lipschitz type estimates

KW - Schatten-von Neumann classes

KW - Triple operator integrals

UR - http://www.scopus.com/inward/record.url?scp=84962527889&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2016.02.030

DO - 10.1016/j.aim.2016.02.030

M3 - Article

AN - SCOPUS:84962527889

VL - 295

SP - 1

EP - 52

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -

ID: 87316715