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In this note, we generalize the results in our previous paper on the Casimir operator and Berezin transform, by showing the $(L^2,L^2)$-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 $1\le p\le \infty .$