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Commutator Inequalities via Schur Products

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Operator Algebras and Applications

Part of the book series: Abel Symposia ((ABEL,volume 12))

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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

$$\displaystyle{\|[g(D),y]\| \leq A_{0}\|y\| + A_{1}\|[D,y]\| + A_{2}\|[D,[D,y]]\| +\ldots +A_{n}\|[D,[D,\ldots [D,y]\ldots ]]\|.}$$

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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Author information

Authors and Affiliations

  1. Institute for Mathematics, Universitetsparken 5, 2100, Copenhagen O, Denmark

    Erik Christensen

Authors
  1. Erik Christensen
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Correspondence to Erik Christensen .

Editor information

Editors and Affiliations

  1. Department of Science and Technology, University of the Faroe Islands, Tórshavn, Faroe Islands

    Toke M. Carlsen

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Nadia S. Larsen

  3. Department of Mathematics, University of Oslo, Oslo, Norway

    Sergey Neshveyev

  4. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Christian Skau

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Christensen, E. (2016). Commutator Inequalities via Schur Products. In: Carlsen, T.M., Larsen, N.S., Neshveyev, S., Skau, C. (eds) Operator Algebras and Applications. Abel Symposia, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-39286-8_5

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