Hankel Operators and Schatten—von Neumann Classes
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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 Hφ ∈ S 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 φ ∈ B 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.
KeywordsBounded Operator Rational Approximation Besov Space Weak Type Hankel Operator
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