Berezin transform of products of Toeplitz operators on the Hardy space

Abstract

Let H 2 ( S ) H^2(S) be the Hardy space on the unit sphere in C n \mathbf {C}^n . We show that there are Toeplitz operators T f T_f and T g T_g on H 2 ( S ) H^2(S) such that the product T f T g T_fT_g is not compact and yet ‖ T f T g k z ‖ \|T_fT_gk_z\| tends to 0 0 as | z | → 1 |z| \rightarrow 1 . Consequently, the Berezin transform ⟨ T f T g k z , k z ⟩ \langle T_fT_gk_z,k_z\rangle tends to 0 0 as | z | → 1 |z| \rightarrow 1 .