Product isometry of generalized weighted composition operator on general weighted Hardy space

TL;DR

This paper characterizes when composition and weighted composition operators are isometric or unitary on a generalized weighted Hardy space, including conditions for their boundedness and examples of isometric products.

Contribution

It provides necessary and sufficient conditions for isometry and unitarity of composition operators on generalized weighted Hardy spaces, and characterizes their boundedness.

Findings

Conditions for composition operators to be isometric and unitary.

Counterexamples showing products of non-isometric operators can be isometric.

Complete characterization of boundedness of generalized weighted composition operators.

Abstract

We obtain necessary and sufficient conditions for the composition and weighted composition operator and product of composition operators to be isometry and unitary on $H_{E}(\xi).$ With the help of counter example we also prove that the product of two non isometric composition operator and weighted composition operator can be isometry on $H_{E}(\xi)$. We also completely characterize the boundedness of generalized weighted composition operators on $H_{E}(\xi).$