On Some Problems for Reproducing Kernel Hilbert Space Operators via the Berezin Transform

We give some applications of Berezin transforms and Engli ' s C*-algebras methods, namely we investigate the solution of generalized Riccati operator equation of the form XAX + Sigma(N)(i=1) BiXCi - D = 0, N >= 2 via Berezin transforms. Also, we study finite product of operators, including finite zero-product of Toeplitz operators on the Bergman Hilbert space L-a(2) (D) in terms of Berezin transforms. The same method is used for characterization of compact truncated operators on the reproducing kernel Hilbert space H = H(Omega)