The kernel spaces and Fredholmness of truncated Toeplitz operators

In this paper, we study some conditions about invertible and Fredholm truncated Toeplitz operators which have unique symbols. For f ∈ L ∞ , if Af is a Fredholm operator, then f|E ̸= 0 for any E ⊂ T with |E| > 0. Moreover ind (Af ) = 0. In particular, if Af is invertible in L(K2 u), then f is invertible in L ∞ . Besides, we give some results about the kernel spaces of truncated Toeplitz operators. For f ∈ L ∞ , we obtain the necessary and sufficient condition that the defect operator I − A ∗ fAf of truncated Toeplitz operator Af meeting some conditions is compact on the model space K2 u .

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