Operator algebras generated by left invertibles

TL;DR

This paper investigates the structure of operator algebras generated by left invertible operators and their Moore-Penrose inverses, revealing connections to Toeplitz algebras, Cowen-Douglas operators, and conditions for algebra isomorphism.

Contribution

It characterizes the algebra generated by a left invertible operator and its Moore-Penrose inverse, especially under the analytic and Fredholm index conditions, linking to well-known operator classes.

Findings

T is analytic iff T* is Cowen-Douglas.

For Fredholm index -1, the algebra contains compact operators.

Algebras are isomorphic iff they are similar.

Abstract

Operator algebras generated by partial isometries and their adjoints form the basis for some of the most well studied classes of C*-algebras. The primary object of this paper is the norm-closed operator algebra generated by a left invertible $T$ together with its Moore-Penrose inverse $T^\dagger$. We denote this algebra by $\mathfrak{A}_T$. In the isometric case, $T^\dagger = T^*$ and $\mathfrak{A}_T$ is a representation of the Toeplitz algebra. Of particular interest is the case when $T$ satisfies a non-degeneracy condition called analytic. We show that $T$ is analytic if and only if $T^*$ is Cowen-Douglas. When $T$ is analytic with Fredholm index $-1$, the algebra $\mathfrak{A}_T$ contains the compact operators, and any two such algebras are boundedly isomorphic if and only if they are similar.