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Hilbert-Schmidt Localization Operators

  • M. W. Wong
Part of the Operator Theory: Advances and Applications book series (OT, volume 136)

Abstract

Let π : GU(X) be an irreducible and square-integrable representation of a locally compact and Hausdorff group G on a Hilbert space X. Then for all functions F in L p (G), 1 ≤ p ≤ ∞, and all admissible wavelets φ for π : GU(X), Proposition 12.3 ensures that we can get a unique bounded linear operator L F,φ : XX such that
$$\parallel {{L}_{{F,\varphi }}}{{\parallel }_{*}} \leqslant {{\left( {\frac{1}{{{{c}_{\varphi }}}}} \right)}^{{\tfrac{1}{p}}}}\parallel F{{\parallel }_{{{{L}^{p}}(G)}}}$$
(15.1)
and
$$({{L}_{{F,\varphi }}}x,y) = \frac{1}{{{{c}_{\varphi }}}}\int_{G} {F(g)(x,\pi (g)\varphi )(\pi (g)\varphi ,y)d\mu (g)}$$
(15.2)
for all x and y in X whenever F is a simple function on G for which
$$\mu \{ g \in G:F(g) \ne 0\} < \infty .$$

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Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • M. W. Wong
    • 1
  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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