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Hankel Operators and Schatten—von Neumann Classes

  • Vladimir Peller
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter we study Hankel operators that belong to the Schattenvon Neumann class S p , 0 < p < ∞. The main result of the chapter says that HφS p if and only if the function P‒φ, belongs to the Besov class B p 1/p (see Appendix 2.6). We prove this result in §1 for p =1. We give two different approaches. The first approach gives an explicit representation of a Hankel operator in terms of rank one operators while the second approach is less constructive but it allows one to represent a nuclear Hankel operator as an absolutely convergent series of rank one Hankel operators. We also characterize in §1 nuclear Hankel operators of the form Γ[µ] in terms of measures µ, in 𝔻. In §2 we prove the main result for 1 < p < ∞. We use the result for p = 1 and the Marcinkiewicz interpolation theorem for linear operators. Finally, in §3 we treat the case p < 1. To prove the necessity of the condition φB p 1/p , we reduce the estimation of Hankel matrices to the estimation of certain special finite matrices that are normal and whose norms can be computed explicitly.

Keywords

Bounded Operator Rational Approximation Besov Space Weak Type Hankel Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 2003

Authors and Affiliations

  • Vladimir Peller
    • 1
  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA

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