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

Abstract

Abstract Let D ${\mathbb {D}} $ be the open unit disk of the complex plane C ${\mathbb {C}} $ and H(D) $H({\mathbb {D}} )$ be the space of all analytic functions on D ${\mathbb {D}} $. Let Aγ,δ2(D) $A^{2}_{\gamma ,\delta }({\mathbb {D}} )$ be the space of analytic functions that are L2 $L^{2}$ with respect to the weight ωγ,δ(z)=(ln1|z|)γ[ln(1−1ln|z|)]δ $\omega _{\gamma ,\delta }(z)=( \ln \frac{1}{|z|})^{\gamma }[\ln (1-\frac{1}{\ln |z|})]^{\delta }$, where −1<γ<∞ $-1<\gamma <\infty $ and δ≤0 $\delta \le 0$. For given g∈H(D) $g\in H({\mathbb {D}} )$, the integral-type operator Ig $I_{g}$ on H(D) $H({\mathbb {D}} )$ is defined as Igf(z)=∫0zf(ζ)g(ζ)dζ. $$ I_{g}f(z)= \int _{0}^{z}f(\zeta )g(\zeta )\,d\zeta . $$ In this paper, we characterize the boundedness of Ig $I_{g}$ on Aγ,δ2 $A^{2}_{\gamma ,\delta }$, whereas in the main result we estimate the essential norm of the operator. Some basic results on the space Aγ,δ2(D) $A^{2}_{\gamma ,\delta }({\mathbb {D}} )$ are also presented.

Keywords

• Integral-type operator
• Boundedness
• Compactness
• Essential norm