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 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.