Boundedness and essential norm of an integral-type operator on a Hilbert–Bergman-type spaces

Abstract

Let D be the open unit disk of the complex plane C and H(D) be the space of all analytic functions on D. Let Aγ,δ2(D) be the space of analytic functions that are L 2 with respect to the weight ωγ,δ(z)=(ln1|z|)γ[ln(1−1ln|z|)]δ, where − 1 < γ< ∞ and δ≤ 0. For given g∈ H(D) , the integral-type operator I g on H(D) is defined as Igf(z)=∫0zf(ζ)g(ζ)dζ. In this paper, we characterize the boundedness of I g on Aγ,δ2, whereas in the main result we estimate the essential norm of the operator. Some basic results on the space Aγ,δ2(D) are also presented.