A cauchy problem for elliptic invariant differential operators and continuity of a generalized berezin transform

In this note, we generalize the results in our previous paper on the Casimir operator and Berezin transform, by showing the (L2, L2) -continuity of a generalized Berezin transform associated with a branching problem for a class of unitary representations defined by invariant elliptic operators; we also show, that under suitable general conditions, this generalized Berezin transform is (L-p, L-p) -continuous for 1 <= p <= infinity.