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Perturbation of operators and applications to frame theory

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Abstract

A celebrated classical result states that an operator U on a Banach space is invertible if it is close enough to the identity operator I in the sense that ‖I−U‖<1. Here we show that U actually is invertible under a much weaker condition. As an application we prove new theorems concerning stability offrames (and frame-like decompositions) under perturbation in both Hilbert spaces and Banach spaces.

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, 65211, Columbia, MO

    Peter G. Cazassa

  2. Mathematical Institute, Technical University of Denmark, Building 303, 2800, Lyngby, Denmark

    Ole Christensen

Authors
  1. Peter G. Cazassa
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  2. Ole Christensen
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Additional information

The first named author is partially supported by grants from the U.S. National Science Foundation (grant no. NSF DMS-9201357), the Danish Natural Science Research Council (Grant no. 9401598), and grants from the University of Missouri System Research Board, and the MU Research Council. The second named author thanks the University of Missouri for its hospitality during a visit, where the first draft of the paper was written.

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Cazassa, P.G., Christensen, O. Perturbation of operators and applications to frame theory. The Journal of Fourier Analysis and Applications 3, 543–557 (1997). https://doi.org/10.1007/BF02648883

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  • DOI: https://doi.org/10.1007/BF02648883

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