SOME RESULTS FOR WEIGHTED BERGMAN SPACE OPERATORS VIA BEREZIN SYMBOLS

We consider the weighted Bergman space A(alpha)(2) (D) of analytic functions f on the unit disc D = {z is an element of C : |z| < 1} such that parallel to f parallel to(2)(A alpha 2) :=integral(D) |f(z)|(2) dA(alpha) (z) < +infinity, where d A(alpha)(z) = (alpha+1) (1 - |z|(2))(alpha) d A(z) and dA(z) = dxdy/pi (the normalized Lebesgue area measure). We investigate the generalized Riccati operator equation XA(1)X + A(2)XA(3) + A(4)YA(5) + A(6) = 0 with A(i) is an element of B(A(alpha)(2) (D) ), i = (1,6) over bar. Also we study the solvability of the operator equation X1T phi 1 + X2T phi 2 + center dot center dot center dot + XnT phi n = I-A alpha 2, in the set of Toeplitz operators on the weighted Bergman space A(alpha)(2) (D). Moreover, we characterize compactness of the operator T phi T psi - T-eta in terms of Berezin transforms. (c) 2024 Mathematical Institute Slovak Academy of Sciences