The kernel spaces and Fredholmness of truncated Toeplitz operators

Abstract

In this paper, we study some conditions about invertible and Fredholm truncated Toeplitz operators which have unique symbols. For $f\in L^\infty$, if $A_f$ is a Fredholm operator, then $f _E\neq 0$ for any $E\subset \mathbb{T}$ with $ E >0$. Moreover \textnormal {ind} $(A_f)=0.$ In particular, if $A_f$ is invertible in $\mathfrak{L}(K_u^2)$, then $f$ is invertible in $L^\infty$. Besides, we give some results about the kernel spaces of truncated Toeplitz operators. For $f \in L^\infty$, we obtain the necessary and sufficient condition that the defect operator $I-A_f^*A_f$ of truncated Toeplitz operator $A_f$ meeting some conditions is compact on the model space $K_u^2$.