Fredholmness of Toeplitz operators and corona problems

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 φM+ = φ M- 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.