Using techniques from the theory of reproducing kernels and Berezin symbols, we investigate some problems related to classes of linear operators acting on reproducing kernel Hilbert spaces (RKHS's). In particular, we establish new estimates related to the numerical radii and Berezin numbers of some operators on RKHS's. Further, in terms of the distance function, we describe invariant subspaces of isometric composition operators on a RKHS H(Ω) of complex-valued, but not necessarily analytic, functions on a set Ω. Moreover, we consider a modification of Sarason's question about truncated Toeplitz operators. We also discuss related problems.

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project no. RGP-VPP-323.