Invertibility of Toeplitz operators and corona conditions in a strip

A Toeplitz operator with symbol G such that det G = 1 is invertible if there is a non-trivial solution to a Riemann-Hilbert problem G phi(+) = phi(-) with phi(+) and phi(-) satisfying the corona conditions in C+ and C-, respectively. However, determining such a solution and verifying that the corona conditions are satisfied are in general difficult problems. In this paper, on one hand, we establish conditions on phi(+/-) which are equivalent to the corona conditions but easier to verify, if G(+/- 1) are analytic and bounded in a strip. This happens in particular with almost-periodic symbols. On the other hand, we identify new classes of symbols G for which a non-trivial solution to G phi(+) = phi(-) can be explicitly determined and the corona conditions can be verified by the above mentioned approach, thus obtaining invertibility criteria for the associated Toeplitz operators. (c) 2007 Elsevier Inc. All rights reserved.