Certain nontrivially essentially self-adjoint weighted composition operators on H2 and Aα2

In this paper, we characterize nontrivially essentially self-adjoint weighted composition operators C-psi,C- phi on the Hardy space H-2 and the weighted Bergman spaces A(alpha)(2), when phi is an element of LFT(D) is not an automorphism and psi is an element of H-infinity is continuous at F(phi). Moreover, we show that weighted composition operators C-psi,C- phi on H-2 and A(alpha)(2) are not nontrivially essentially self-adjoint, whenever psi is an element of A(D) and phi(z) = (a - z)/(1 - (a) over barz) for a is an element of D.