Commutator inequalities via Schur products

Erik Christensen

doi:10.1007/978-3-319-39286-8

Commutator inequalities via Schur products

Erik Christensen

Department of Mathematical Sciences

1 Citation (Scopus)

Abstract

For a self-adjoint unbounded operator D on a Hilbert space H; a bounded operator y on H and some Borel functions g(t) we establish inequalities of the type∥[g(D), y] ∥ ≤ A₀∥y∥+1∥[D, y] ∥+A₂∥[D, [D, y]] ∥++A_n∥[D, [D,. [D, y].]] ∥ ∥. The proofs take place in a space of infinite matrices with operator entries, and in this setting it is possible to approximate the matrix associated to [g(D), y] by the Schur product of a matrix approximating [D, y] and a scalar matrix. A classical inequality on the norm of Schur products may then be applied to obtain the results.

Original language	English
Title of host publication	Operator Algebras and Applications : The Abel Symposium 2015
Editors	Toke M. Carlsen, Nadia S. Larsen, Sergey Neshveyev, Christian Skau
Publisher	Springer
Publication date	2016
Pages	127-143
ISBN (Print)	978-3-319-39284-4
ISBN (Electronic)	978-3-319-39286-8
DOIs	https://doi.org/10.1007/978-3-319-39286-8
Publication status	Published - 2016
Event	Abel Symposium 2015 - Bergen/Harstad, Norway Duration: 7 Aug 2016 → 11 Aug 2016

Conference

Conference	Abel Symposium 2015
Country/Territory	Norway
City	Bergen/Harstad
Period	07/08/2016 → 11/08/2016

Series	Abel Symposia
Volume	12

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Commutator inequalities via Schur products. / Christensen, Erik.

Operator Algebras and Applications: The Abel Symposium 2015. ed. / Toke M. Carlsen; Nadia S. Larsen; Sergey Neshveyev; Christian Skau. Springer, 2016. p. 127-143 (Abel Symposia, Vol. 12).

Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review

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abstract = "For a self-adjoint unbounded operator D on a Hilbert space H; a bounded operator y on H and some Borel functions g(t) we establish inequalities of the type∥[g(D), y] ∥ ≤ A0∥y∥+1∥[D, y] ∥+A2∥[D, [D, y]] ∥++An∥[D, [D,. [D, y].]] ∥ ∥. The proofs take place in a space of infinite matrices with operator entries, and in this setting it is possible to approximate the matrix associated to [g(D), y] by the Schur product of a matrix approximating [D, y] and a scalar matrix. A classical inequality on the norm of Schur products may then be applied to obtain the results.",

author = "Erik Christensen",

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editor = "Carlsen, {Toke M. } and Larsen, {Nadia S. } and Neshveyev, {Sergey } and Skau, {Christian }",

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N2 - For a self-adjoint unbounded operator D on a Hilbert space H; a bounded operator y on H and some Borel functions g(t) we establish inequalities of the type∥[g(D), y] ∥ ≤ A0∥y∥+1∥[D, y] ∥+A2∥[D, [D, y]] ∥++An∥[D, [D,. [D, y].]] ∥ ∥. The proofs take place in a space of infinite matrices with operator entries, and in this setting it is possible to approximate the matrix associated to [g(D), y] by the Schur product of a matrix approximating [D, y] and a scalar matrix. A classical inequality on the norm of Schur products may then be applied to obtain the results.

AB - For a self-adjoint unbounded operator D on a Hilbert space H; a bounded operator y on H and some Borel functions g(t) we establish inequalities of the type∥[g(D), y] ∥ ≤ A0∥y∥+1∥[D, y] ∥+A2∥[D, [D, y]] ∥++An∥[D, [D,. [D, y].]] ∥ ∥. The proofs take place in a space of infinite matrices with operator entries, and in this setting it is possible to approximate the matrix associated to [g(D), y] by the Schur product of a matrix approximating [D, y] and a scalar matrix. A classical inequality on the norm of Schur products may then be applied to obtain the results.

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Commutator inequalities via Schur products

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