Abstract
In this study, some problems of operator theory on the reproducing kernel Hilbert space by using the Berezin symbols method are investigated. Namely, invariant subspaces of weighted composition operators on and of operators and A on the reproducing kernel Hilbert space are given. Also, reverse inequalities for the Berezin number of two operators are proved. Under some conditions we prove the power inequality
ber (A(n)) <= (ber (A))(n)
which is related to a well known analogue estimate of Halmos for the numerical radius.