Abstract
We study some problems of operator theory by using this bounded function A similar to , including estimates for Berezin numbers of some operators, including truncated Toeplitz operators. We also prove an operator analog of some Young inequality and use it in proving of some inequalities for Berezin number of operators including the inequality ber (AB) < ber (A) ber (B) , for some operators A and B on 74 (1 >=). More-over, we give in terms of the Berezin number a necessary condition for hyponormality of some operators.