Paraproducts and Continuity of Hankel Operators of Schatten- von Neumann p- Classes and Lorentz Ideals

Abstract

We give an interpolation-free proof of the known fact that a dyadic paraproduct is of Schatten–von Neumann class S_p; if and only if its symbol is in the dyadic Besov space. We use the same technique to prove a corresponding result for dyadic paraproducts with operator symbols. We investigate Hankel operators with anti-holomorphic symbols, on general Fock spaces. For polynomial symbols we will give necessary and sufficient conditions for continuity and compactness a complete characterization of the Schatten–von Neumann p-class membership. We show that the closure of holomorphic polynomials in Hilbert space is a reproducing Kernel Hilbert space of analytic functions and describe various spectral properties of the corresponding Hankel operators with anti- holomorphic symbols. We show the membership of Hankel operator in Lorentz ideals classes