# Wold decompositions for operators close to isometries

## TL;DR

This paper extends the Wold decomposition to certain left invertible operators in Bergman and Dirichlet spaces, providing conditions under which such a decomposition is possible.

## Contribution

It introduces conditions for Wold decompositions of operators close to isometries, specifically for left invertible operators in specific function spaces.

## Key findings

- Established conditions for Wold decompositions in Bergman and Dirichlet spaces.
- Extended classical Wold decomposition to non-isometric operators.
- Provided operator-theoretic criteria for decompositions in function spaces.

## Abstract

In Bergman and Dirichlet spaces, the shift operator is not an isometry, but it is a left invertible operator. In this paper we give conditions on the left invertible operators such that a operator version, in the sense of Rosenblum and Rovnyak, of the Wold decomposition to take place.