Criteria for Toeplitz operators on the sphere

Abstract:

Let $H^{2}(S)$ be the Hardy space on the unit sphere $S$ in $\mathbf {C}^{n}$. We show that a set of inner functions $\Lambda$ is sufficient for the purpose of determining which $A\in {\mathcal {B}}(H^{2}(S))$ is a Toeplitz operator if and only if the multiplication operators $\{M_{u} : u \in \Lambda \}$ on $L^{2}(S,d\sigma )$ generate the von Neumann algebra $\{M_{f} : f \in L^{\infty }(S,d\sigma )\}$.