Description of Bloch spaces, weighted Bergman spaces and invariant subspaces, and related questions

Let D be the unit disc of complex plane C, and H = Hol(D) the class of functions analytic in D. Recall that an f is an element of Hol(D) is said to belong to the Bloch space B=B(D) if parallel to f parallel to(B) :=sup(z is an element of D)(1-vertical bar z vertical bar(2))vertical bar f'(z)vertical bar<+infinity. With the norm parallel to f parallel to =vertical bar f(0)vertical bar+parallel to f parallel to(B), B is Banach space. Let B-0 = B-0(D)be the Bloch space which consists of all f is an element of B satisfying lim(vertical bar z vertical bar -> 1)(1-vertical bar z vertical bar(2))vertical bar f'(z)vertical bar=0. Here we give a new description of Bloch spaces and weighted Bergman spaces in terms of Berezin symbols of diagonal operators associated with the Taylor coefficients of their functions. We also give in terms of Berezin symbols a characterization of the multiple shift invariant subspaces of weighted Bergman spaces. Some other questions are also discussed.