Schatten class membership of Hankel operators on the unit sphere

Let H2(S) be the Hardy space on the unit sphere S in Cn, n 2. Consider the Hankel operator Hf = (1−P)Mf |H2(S), where the symbol function f is allowed to be arbitrary in L2(S, dσ).We show that for p >2n, Hf is in the Schatten class Cp if and only if f − Pf belongs to the Besov space Bp. To be more precise, the “if” part of this statement is easy. The main result of the paper is the “only if” part. We also show that the membership Hf ∈ C2n implies f −Pf = 0, i.e., Hf = 0. © 2009 Elsevier Inc. All rights reserved