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.

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