Abstract
We study the Fredholm properties of Toeplitz operators with bounded symbols of vanishing mean oscillation in the complex plane. In particular, we prove that the Toeplitz operator with such a symbol is Fredholm on a standard weighted Fock space if and only if the Berezin transform of the symbol is bounded away from zero outside a sufficiently large disk in the complex plane. We also show that the Fredholm index of the Toeplitz operator can be computed via the winding of the symbol along a sufficiently large circle. We finish by considering Toeplitz operators with matrix-valued symbols.
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