Some results for reproducing kernel Hilbert space operators via Berezin symbols

Abstract

By applying the Berezin symbols method, we investigate the solvability of the Riccati operator equation XAX+XB-CX-D=0 on the set of operators of the form Toeplitz + compact on the Bergman space L_{a}²(D) of analytic functions in the unit disc D={z∈C:|z|<1}. We also characterize compact truncated operators on the standard reproducing kernel Hilbert space in the sense of Nordgren and Rosenthal. Moreover, we discuss solvability of the equation

T_{ϕ₁}X₁+T_{ϕ₂}X₂+...+T_{ϕ_{n}}X_{n}=I+K,

where T_{ϕ_{i}} (i=1,n) is the Toeplitz operator on L_{a}²(D) and K:L_{a}²(D)→L_{a}²(D) is a fixed compact operator.