Reproducing kernels, Englis algebras and some applications

We introduce the notion of Englis algebras, defined in terms of reproducing kernels and Berezin symbols. Such algebras were apparently first investigated by Englis (1995). Here we give some new results on Englis C*-algebras on abstract reproducing kernel Hilbert spaces and some applications to various questions of operator theory. In particular, we give applications to Riccati operator equations, zero Toeplitz products, and the existence of invariant subspaces for some operators.