Generalize Hilbert operator acting on Dirichlet spaces

TL;DR

This paper characterizes measures for which a generalized Hilbert operator, acting on Dirichlet spaces, is bounded or compact, extending classical operator theory results to a broader class of function spaces.

Contribution

It provides new characterizations of measures ensuring the boundedness and compactness of a generalized Hilbert operator between specific Dirichlet spaces.

Findings

Characterization of measures for boundedness of the operator

Criteria for compactness of the operator

Extension of classical results to generalized settings

Abstract

Let $\mu$ be a positive Borel measure on the interval $[0,1)$. For $\gamma>0$, the Hankel matrix $\mathcal{H}_{\mu,\gamma}=(\mu_{n,k})_{n,k\geq0}$ with entries $\mu_{n,k}=\mu_{n+k}$, where $\mu_{n+k}=\int_{0}^{\infty}t^{n+k}d\mu(t)$. formally induces the operator $$\mathcal{H}_{\mu,\gamma}=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}\mu_{n,k}a_k\right)\frac{\Gamma(n+\gamma)}{n!\Gamma(\gamma)}z^n,$$ on the space of all analytic functions $f(z)=\sum_{k=0}^{\infty}{a_k}{z^k}$ in the unit disc $\mathbb{D}$. Following ideas from \cite{author3} and \cite{author4}, in this paper, for $0\leq\alpha<2$, $2\leq\beta<4$, $\gamma\geq1$. we characterize the measure $\mu$ for which $\mathcal{H}_{\mu,\gamma}$ is bounded(resp.,compact)from $\mathcal{D}_{\alpha}$ into $\mathcal{D}_{\beta}$.