Reproducing Kernels and Berezin Symbols Techniques in Various Questions of Operator Theory

In this paper we give some applications of reproducing kernels and Berezin symbols techniques in various questions of operator theory in the functional Hilbert spaces of complex-valued functions. In particular, by using these techniques, and also the so-called distance function method of Nikolski, we investigate invariant subspace problem and invertibility problem of operators. We also prove some general unicity theorem connecting with distance function. Moreover we introduce the concepts of Berezin set and Berezin number of operators and study some properties.