# Co-isometric weighted composition operators on Hilbert spaces of analytic functions

## TL;DR

This paper characterizes when weighted composition operators are co-isometric on weighted Hardy spaces, revealing that such operators are unitary and identifying specific spaces that support non-trivial examples.

## Contribution

It provides a necessary and sufficient condition for co-isometric weighted composition operators on weighted Hardy spaces, showing they are equivalent to unitary operators.

## Key findings

- Co-isometric weighted composition operators are equivalent to unitary operators.
- A specific family of weighted Hardy spaces uniquely supports non-trivial co-isometric operators.
- The result establishes a dichotomy in the structure of these operators across different spaces.

## Abstract

We obtain a necessary and sufficient condition for a weighted composition operator to be co-isometric on a general weighted Hardy space of analytic functions in the unit disk whose reproducing kernel has the usual natural form. This turns out to be equivalent to the property of being unitary. The result reveals a dichotomy identifying a specific family of weighted Hardy spaces as the only ones that support non-trivial operators of this kind.