A counterexample of the Fredholm of Toeplitz operator

TL;DR

The paper creates a concrete continuous symbol $a(z)=\frac{1-\ln(1+z)}{1-\ln(1+z^{-1})}$ to disprove Virtanen's conjecture that $Qa$ in $\mathbf{BMO}_{\log}$ ensures Fredholmness of the Toeplitz operator $T_a$ on $H^1(\mathbb{T})$. It develops a detailed $\mathbf{BMO}_{\log}$ analysis by showing $Pa$ and $Qa$ lie in $\mathbf{BMO}_{\log}$ via careful control of $\ln a^{+}$ and $\ln a^{-}$ and a precise study of their singularity at $-1$. The authors establish $\ker T_a=\{0\}$ and that $R(T_a)$ is dense in $H^1(\mathbb{T})$, then construct a function in $H^1(\mathbb{T})$ with no preimage on a boundary arc, concluding that $R(T_a)$ is closed, hence $T_a$ is not Fredholm. This provides a counterexample to the expectation that $Qa\in \mathbf{BMO}_{\log}$ suffices for Fredholmness, clarifying the limits of Fredholm theory for Toeplitz operators on $H^1$. The result has implications for understanding boundary behavior and spectral properties of Toeplitz operators in the $p=1$ Hardy space regime.

Abstract

In this paper we study the essential spectra of the Toeplitz operator on the Hardy space $H^1$. We give a counterexample to show that the Toeplitz operator with symbol is not Fredholm, which gives a counterexample to the conjecture by J.A. Virtanen J A in 2006.

Theorems & Definitions (21)

• Example 2.1
• Example 2.2
• Lemma 2.3
• proof
• Definition 2.4
• Lemma 3.1
• proof
• Corollary 3.2
• proof
• Proposition 3.3