Schatten class membership of Hankel operators on the unit sphere

Abstract

Let H 2 ( S ) be the Hardy space on the unit sphere S in C n , n ⩾ 2 . Consider the Hankel operator H f = ( 1 − P ) M f | H 2 ( S ) , where the symbol function f is allowed to be arbitrary in L 2 ( S , d σ ) . We show that for p > 2 n , H f is in the Schatten class C p if and only if f − P f belongs to the Besov space B p . 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 H f ∈ C 2 n implies f − P f = 0 , i.e., H f = 0 .