BMO IN THE BERGMAN METRIC ON BOUNDED SYMMETRICAL DOMAINS

For bounded symmetric domains Ω in Cn, a notion of "bounded mean oscillation" in terms of the Bergman metric is introduced. It is shown that for f{hook} in L2(Ω, dv), f{hook} is in BMO(Ω) if and only if the densely-defined operator [Mf{hook}, P] ≡ Mf{hook}P - PMf{hook} on L2(Ω, dv) is bounded (here, Mf{hook} is "multiplication by f{hook}" and P is the Bergman projection with range the Bergman subspace H2(Ω, dv) = La2(Ω, dv) of holomorphic functions in L2(Ω, dv)). An analogous characterization of compactness for [Mf{hook}, P] is provided by functions of "vanishing mean oscillation at the boundary of Ω". © 1990.