Fredholm theory of Toeplitz operators on the Hardy space H1

The Fredholm properties of Toeplitz operators T-a on Hardy spaces H-p (1 < p < infinity) with continuous symbols alpha are well understood. We consider T-a acting on H-1, where the operator is bounded provided that alpha belongs to the class of symbols given by Janson and Stegenga's result on the pointwise multipliers on H-1. A necessary and sufficient condition for T-a to be a Fredholm operator is given when alpha is continuous and satisfies a mild additional condition (much weaker than Holder continuity). A formula for the index of T-a is also derived. In addition, we study the case of matrix-valued symbols and Toeplitz operators on BMOA