Generalized Hilbert operators acting on weighted spaces of holomorphic functions with sup-norms

TL;DR

This paper investigates the properties of generalized Hilbert operators on weighted holomorphic function spaces, establishing conditions for their boundedness, compactness, and nuclearity, with notable differences from Cesàro-type operators.

Contribution

It provides new characterizations of boundedness, compactness, and nuclearity of generalized Hilbert operators on various weighted holomorphic spaces, including classical Hardy and Wiener spaces.

Findings

Boundedness and compactness are equivalent for certain weights.

Summability of measure moments characterizes operator properties.

Nuclearity is linked to measure moment summability.

Abstract

The behaviour of the generalized Hilbert operator associated with a positive finite Borel measure $\mu$ on $[0,1)$ is investigated when it acts on weighted Banach spaces of holomorphic functions on the unit disc defined by sup-norms and on Korenblum type growth Banach spaces. It is studied when the operator is well defined, bounded and compact. To this aim, we study when it can be represented as an integral operator. We observe important differences with the behaviour of the Ces\`aro-type operator acting on these spaces, getting that boundedness and compactness are equivalent concepts for some standard weights. For the space of bounded holomorphic functions on the disc and for the Wiener algebra, we get also this equivalence, which is characterized in turn by the summability of the moments of the measure $\mu.$ In the latter case, it is also equivalent to nuclearity. Nuclearity of the generalized Hilbert operator acting on related spaces, such as the classical Hardy space, is also analyzed.