Invertibility and Fredholm Property of Fock Toeplitz Operators

We characterize some necessary and sufficient conditions of invertible Toeplitz operators acting on the Fock space. In particular, we study the Fredholm properties of Toeplitz operators with BMO1 symbols, where their Berezin transforms are bounded functions of vanishing oscillation. We show the Fredholm index of the Toeplitz operator via the winding of its Berezin transform along a sufficiently large circle and provide a characterization of invertible Toeplitz operators with non-negative symbols, possibly unbounded, such that the Berezin transforms of the symbols are bounded and of vanishing oscillation.