Invertibility of Toeplitz operators and corona conditions in a strip

A Toeplitz operator with symbol G such that detG=1 is invertible if there is a non-trivial solution to a Riemann–Hilbert problem Gϕ+=ϕ− with ϕ+ and ϕ− 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 ϕ± 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 Gϕ+=ϕ− 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.