# A Gleason-Kahane-\.Zelazko theorem for the Dirichlet space

## TL;DR

This paper extends the Gleason-Kahane-Żelazko theorem to the Dirichlet space, characterizing certain linear functionals and operators without assuming continuity, and explores properties of weighted Dirichlet spaces.

## Contribution

It establishes a Gleason-Kahane-Żelazko type theorem for the Dirichlet space and characterizes weighted composition operators based on their action on nowhere-vanishing functions.

## Key findings

- Linear functionals non-zero on nowhere-vanishing functions are point evaluations.
- Weighted composition operators preserve nowhere-vanishing functions.
- Identification of functions mapping the disk onto the entire complex plane.

## Abstract

We show that every linear functional on the Dirichlet space that is non-zero on nowhere-vanishing functions is necessarily a multiple of a point evaluation. Continuity of the functional is not assumed. As an application, we obtain a characterization of weighted composition operators on the Dirichlet space as being exactly those linear maps that send nowhere-vanishing functions to nowhere-vanishing functions. We also investigate possible extensions to weighted Dirichlet spaces with superharmonic weights. As part of our investigation, we are led to determine which of these spaces contain functions that map the unit disk onto the whole complex plane.