The kernel spaces, Fredholmness and compact defect

In this talk, we study some conditions about invertible and Fredholm truncated Toeplitz operators which have unique symbols. Obviously, if the defect operator of truncated Toeplitz operator $A_f$ is compact, then $A_f$ is a Fredholm operator. In addtion, we provide the necessary and sufficient condition that the defect operator $I-A_f^*A_f$ of truncated Toeplitz operator $A_f$ for $f\in H^\infty$ with $\|f\|_\infty\leq1$ is of finite-rank on the model space $K_u^2$. Moreover, the necessary and sufficient condition is obtained for $I-A_f^*A_f$ with $f\in K_u^2\cap L^\infty$ to be compact on the model space $K_u^2$. For $f\in L^\infty$, we get 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 $K_u^2$. Besides, we give some results about the kernel spaces of truncated Toeplitz operators.