Integral representation of radial operators on the Bergman space over the unit disc

Abstract

Let D be the unit disc in C. In this article, we characterize all analytic functions g on D so that the integral operator Sg defined formally on A2(D) by(Sgf)(z)=∫Df(w)g(zw‾)dA(w),∀f∈A2(D),z∈D is bounded, where dA(w) is the normalized area measure on D. We show that the collection of all bounded operators Sg is exclusively the collection of all radial operators on A2(D). Further, we study several operator theoretic properties of these operators. Also, we give integral representation of the form Sg for all operators in the C⁎-algebra generated by Toeplitz operators Ta with radial symbols a∈L∞(D).