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 epsilon (L-infinity(R))(2x2) is shown to be equivalent to that of a Toeplitz operator with scalar symbol gamma := det G, provided that the Riemann-Hilbert problem G phi(M)(+) = phi(M)(-) admits a solution such that the meromorphic corona problems with data phi(M)(+/-) are solvable. The Fredholm properties are characterized in terms of phi(M)(+/-) and the corresponding meromorphic left-inverses. Partial index estimates for the symbols and Fredholmness criteria arc established for several classes of Toeplitz operators. (C) 2010 Elsevier Inc. All rights reserved.