Invertibility, Fredholmness and kernels of dual truncated Toeplitz operators

TL;DR

This paper introduces and analyzes asymmetric dual truncated Toeplitz operators, exploring their invertibility, Fredholm properties, and spectra, and establishing connections with Carleson's corona theorem.

Contribution

It defines asymmetric dual truncated Toeplitz operators, relates them to paired operators and inverse symbols, and studies their Fredholmness and spectral properties.

Findings

Operators are equivalent after extension to paired operators on $L^2(T)$

Invertibility in $L^(T)$ relates to inverse symbols

Results connect operator properties with Carleson's corona theorem

Abstract

Asymmetric dual truncated Toeplitz operators acting between the orthogonal complements of two (eventually different) model spaces are introduced and studied. They are shown to be equivalent after extension to paired operators on $L^2(\mathbb T) \oplus L^2(\mathbb T)$ and, if their symbols are invertible in $L^\infty(\mathbb T)$, to asymmetric truncated Toeplitz operators with the inverse symbol. Relations with Carleson's corona theorem are also established. These results are used to study the Fredholmness, the invertibility and the spectra of various classes of dual truncated Toeplitz operators.