Continuity and Schatten–von Neumann p-class membership of Hankel operators with anti-holomorphic symbols on (generalized) Fock spaces

Abstract

In this paper we investigate Hankel operators H f ¯ : A m 2 → A m 2 ⊥ with anti-holomorphic symbols f ¯ = ∑ k = 0 ∞ b k z ¯ k ∈ L 2 ( C , | z | m ) , where A m 2 are general Fock spaces. We will show that H f ¯ is not continuous if the corresponding symbol is not a polynomial f ¯ = ∑ k = 0 N b k z ¯ k . For polynomial symbols we will give necessary and sufficient conditions for continuity and compactness in terms of N and m. For monomials z ¯ k we will give a complete characterization of the Schatten–von Neumann p-class membership for p > 0 . Namely in case 2 k < m the Hankel operators H z ¯ k are in the Schatten–von Neumann p-class iff p > 2 m / ( m − 2 k ) ; and in case 2 k ⩾ m they are not in the Schatten–von Neumann p-class.