COMPACT TOEPLITZ OPERATORS WITH CONTINUOUS SYMBOLS

Abstract

AbstractFor any rotation-invariant positive regular Borel measure ν on the closed unit ball $\overline{\mathbb{B}}_n$ whose support contains the unit sphere $\mathbb{S}_n$, let L2a be the closure in L2 = L2($\overline{\mathbb{B}}_n, dν) of all analytic polynomials. For a bounded Borel function f on $\overline{\mathbb{B}}_n$, the Toeplitz operator Tf is defined by Tf(ϕ) = P(fϕ) for ϕ ∈ L2a, where P is the orthogonal projection from L2 onto L2a. We show that if f is continuous on $\overline{\mathbb{B}}_n$, then Tf is compact if and only if f(z) = 0 for all z on the unit sphere. This is well known when L2a is replaced by the classical Bergman or Hardy space.