# Operators invariant relative to a completely nonunitary contraction

## TL;DR

This paper investigates operators invariant under a contraction on a Hilbert space, showing that in a specific case, such operators are compressions of unbounded transformations commuting with the dilation, extending prior results on truncated Toeplitz operators.

## Contribution

It generalizes Sarason's characterization of invariant operators to contractions with equal defect indices of 1, linking them to unbounded commuting transformations.

## Key findings

- Every A-invariant operator is a compression of an unbounded operator commuting with A's dilation.
- Extension of Sarason's results beyond class C_{00} contractions.
- Adaptation of truncated Toeplitz operator properties to a broader context.

## Abstract

Given a contraction A on a Hilbert space H, an operator T on H is said to be A-invariant if <Tx,x>=<TAx,Ax> for every x in H such that ||Ax||=||x||. In the special case in which both defect indices of A are equal to 1, we show that every A-invariant operator is the compression to H of an unbounded linear transformation that commutes with the minimal unitary dilation of A. This result was proved by Sarason under the additional hypothesis that A is of class C_{00}, leading to an intrinsic characterization of the truncated Toeplitz operators. We also adapt to our more general context other results about truncated Toeplitz operators.