On some problems for operators on the reproducing kernel Hilbert space.
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 H2 are studied. Moreover, some new inequalities for the Berezin number of operators are proved. In particular, new reverse inequalities for the Berezin numbers ber(|A|2) and ber (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 ber(A2 ) ≤ (ber (A))n which is related to a well known analogue estimate of Halmos for the numerical radius.
In this paper we use the Duhamel product to provide a Banach algebra structure to each of a scale of Bergman spaces over the unit disk, and then carry out many interesting consequences.
We characterize skew-symmetric operators on a reproducing kernel Hilbert space in terms of their
Berezin symbols. The solution of some operator equations with skew-symmetric operators is…