Abstract
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 are studied. Moreover, some new inequalities for the Berezin number of operators are proved. In particular, new reverse inequalities for the Berezin numbers 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 which is related to a well known analogue estimate of Halmos for the numerical radius.
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