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Functions of perturbed operators. / Peller, Vladimir; Александров, Алексей Борисович.

In: Comptes Rendus Mathematique, Vol. 347, No. 9-10, 05.2009, p. 483-488.

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

Harvard

Peller, V & Александров, АБ 2009, 'Functions of perturbed operators', Comptes Rendus Mathematique, vol. 347, no. 9-10, pp. 483-488. https://doi.org/10.1016/j.crma.2009.03.004

APA

Peller, V., & Александров, А. Б. (2009). Functions of perturbed operators. Comptes Rendus Mathematique, 347(9-10), 483-488. https://doi.org/10.1016/j.crma.2009.03.004

Vancouver

Peller V, Александров АБ. Functions of perturbed operators. Comptes Rendus Mathematique. 2009 May;347(9-10):483-488. https://doi.org/10.1016/j.crma.2009.03.004

Author

Peller, Vladimir ; Александров, Алексей Борисович. / Functions of perturbed operators. In: Comptes Rendus Mathematique. 2009 ; Vol. 347, No. 9-10. pp. 483-488.

BibTeX

@article{328364ee02f947a18070aef1e9f88713,
title = "Functions of perturbed operators",
abstract = "We prove that if 0 < α < 1 and f is in the H{\"o}lder class Λα (R), then for arbitrary selfadjoint operators A and B with bounded A - B, the operator f (A) - f (B) is bounded and {norm of matrix} f (A) - f (B) {norm of matrix} ≤ const {norm of matrix} A - B {norm of matrix}α. We prove a similar result for functions f of the Zygmund class Λ1 (R): {norm of matrix} f (A + K) - 2 f (A) + f (A - K) {norm of matrix} ≤ const {norm of matrix} K {norm of matrix}, where A and K are selfadjoint operators. Similar results also hold for all H{\"o}lder-Zygmund classes Λα (R), α > 0. We also study properties of the operators f (A) - f (B) for f ∈ Λα (R) and selfadjoint operators A and B such that A - B belongs to the Schatten-von Neumann class Sp. We consider the same problem for higher order differences. Similar results also hold for unitary operators and for contractions. To cite this article: A. Aleksandrov, V. Peller, C. R. Acad. Sci. Paris, Ser. I 347 (2009).",
author = "Vladimir Peller and Александров, {Алексей Борисович}",
year = "2009",
month = may,
doi = "10.1016/j.crma.2009.03.004",
language = "English",
volume = "347",
pages = "483--488",
journal = "Comptes Rendus Mathematique",
issn = "1631-073X",
publisher = "Elsevier",
number = "9-10",

}

RIS

TY - JOUR

T1 - Functions of perturbed operators

AU - Peller, Vladimir

AU - Александров, Алексей Борисович

PY - 2009/5

Y1 - 2009/5

N2 - We prove that if 0 < α < 1 and f is in the Hölder class Λα (R), then for arbitrary selfadjoint operators A and B with bounded A - B, the operator f (A) - f (B) is bounded and {norm of matrix} f (A) - f (B) {norm of matrix} ≤ const {norm of matrix} A - B {norm of matrix}α. We prove a similar result for functions f of the Zygmund class Λ1 (R): {norm of matrix} f (A + K) - 2 f (A) + f (A - K) {norm of matrix} ≤ const {norm of matrix} K {norm of matrix}, where A and K are selfadjoint operators. Similar results also hold for all Hölder-Zygmund classes Λα (R), α > 0. We also study properties of the operators f (A) - f (B) for f ∈ Λα (R) and selfadjoint operators A and B such that A - B belongs to the Schatten-von Neumann class Sp. We consider the same problem for higher order differences. Similar results also hold for unitary operators and for contractions. To cite this article: A. Aleksandrov, V. Peller, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

AB - We prove that if 0 < α < 1 and f is in the Hölder class Λα (R), then for arbitrary selfadjoint operators A and B with bounded A - B, the operator f (A) - f (B) is bounded and {norm of matrix} f (A) - f (B) {norm of matrix} ≤ const {norm of matrix} A - B {norm of matrix}α. We prove a similar result for functions f of the Zygmund class Λ1 (R): {norm of matrix} f (A + K) - 2 f (A) + f (A - K) {norm of matrix} ≤ const {norm of matrix} K {norm of matrix}, where A and K are selfadjoint operators. Similar results also hold for all Hölder-Zygmund classes Λα (R), α > 0. We also study properties of the operators f (A) - f (B) for f ∈ Λα (R) and selfadjoint operators A and B such that A - B belongs to the Schatten-von Neumann class Sp. We consider the same problem for higher order differences. Similar results also hold for unitary operators and for contractions. To cite this article: A. Aleksandrov, V. Peller, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

UR - http://www.scopus.com/inward/record.url?scp=64749089711&partnerID=8YFLogxK

U2 - 10.1016/j.crma.2009.03.004

DO - 10.1016/j.crma.2009.03.004

M3 - Article

VL - 347

SP - 483

EP - 488

JO - Comptes Rendus Mathematique

JF - Comptes Rendus Mathematique

SN - 1631-073X

IS - 9-10

ER -

ID: 5209600