Fredholm index of Toeplitz pairs with $H^{\infty}$ symbols

TL;DR

This paper characterizes when Toeplitz pairs with bounded holomorphic symbols on Hardy space over the bidisk are Fredholm and derives an index formula, relying on solving the Corona Problem in $L^p$ spaces.

Contribution

It provides a new characterization of Fredholmness and an explicit index formula for Toeplitz pairs with $H^{{infty}}$ symbols on the bidisk.

Findings

Fredholmness characterized via $L^p$ Corona Problem solutions

Derived explicit index formula for Toeplitz pairs

Established conditions for Fredholmness on Hardy space over the bidisk

Abstract

In the present paper, we characterize the Fredholmness of Toeplitz pairs on Hardy space over the bidisk with the bounded holomorphic symbols, and hence we obtain the index formula for such Toeplitz pairs. The key to obtain the Fredholmness of such Toeplitz pairs is the $L^p$ solution of Corona Problem over $\mathbb{D}^2$.