On some problems for operators on the reproducing kernel Hilbert space. Issue 11 (18th August 2021) - Garayev, M. T.; Guediri, H.; Gürdal, M.; Alsahli, G. M.
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 H 2 are studied. Moreover, some new inequalities for the Berezin number of operators are proved. In particular, new reverse inequalities for the Berezin numbers b e r A 2 and b e r A of operators A 2 = A ∗ A 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 1 b e r A n ≤ b e r A n which is related to a well known analogue estimate of Halmos for the numerical radius.