Abstract
A meromorphic analogue to the corona problem is formulated and studied and its solutions are characterized as being left-invertible in a space of meromorphic functions. The Fredholmness of Toeplitz operators with symbol G ∈ ( L ∞ ( R ) ) 2 × 2 is shown to be equivalent to that of a Toeplitz operator with scalar symbol γ : = det G , provided that the Riemann–Hilbert problem G ϕ + M = ϕ − M admits a solution such that the meromorphic corona problems with data ϕ ± M are solvable. The Fredholm properties are characterized in terms of ϕ ± M and the corresponding meromorphic left-inverses. Partial index estimates for the symbols and Fredholmness criteria are established for several classes of Toeplitz operators.
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