Generalized Hilbert operators acting on weighted sequence spaces

TL;DR

This work introduces a generalized Hilbert operator $H_{\mu}^{\alpha,\beta}$ induced by a finite positive Borel measure $\mu$ on $(0,1)$ and analyzes its boundedness on weighted sequence spaces $l_{w}^{p}$ and their $\infty$-variants. The authors derive necessary and sufficient conditions for boundedness in terms of the integral criteria $\mathcal{C}_{\mu}(\beta,p)$ and $\mathcal{C}_{\mu}(\beta,\infty)$, and prove that the operator norms exactly equal these criteria. They develop a robust framework of kernel identities, generating functions, and carefully constructed test sequences to establish both upper and lower bounds, yielding precise norm equalities. The results extend previous work on (generalized) Hilbert operators and connect to analytic function spaces via corresponding operators on Hardy and Dirichlet-type spaces, providing sharp criteria and norms for boundedness. Overall, the paper offers a complete characterization of when $H_{\mu}^{\alpha,\beta}$ acts boundedly between weighted sequence spaces and clarifies its norm, with implications for related operator theory in analytic function spaces.

Abstract

In this paper we introduce and study a new kind of generalized Hilbert operators, induced by a finite positive Borel measure on (0,1), acting on weighted sequence spaces. We establish a sufficient and necessary condition for the boundedness of these operators. These results extend some related ones obtained recently in [Bull London Math Soc, 55 (2023), no. 6, 2598-2610].

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Remark 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof