Fredholmness of Toeplitz operators and corona problems

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 := detG, provided that the Riemann–Hilbert problem GφM + = φM − admits a solution such that the meromorphic corona problems with data φM ± are solvable. The Fredholm properties are characterized in terms of φM ± and the corresponding meromorphic left-inverses. Partial index estimates for the symbols and Fredholmness criteria are established for several classes of Toeplitz operators. © 2010 Elsevier Inc. All rights reserved.