Hankel Operators and Their Applications pp 231-301 | Cite as

# Hankel Operators and Schatten—von Neumann Classes

## 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 H*φ* ∈ **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.

## Keywords

Bounded Operator Rational Approximation Besov Space Weak Type Hankel Operator## Preview

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