On the invertibility of restricted Toeplitz operators

摘要

I. Introduction. Let Hm denote the Banach algebra of all bounded analytic functions on the open unit disk D, and let If be the space of all essentially bounded Lebesgue measurable functions on dD. The Banach algebra If is isometrically isomorphic to C(X) via the Gelfand transform, where X = M(If) is the maximal ideal space of If. The Gelfand transform carries H~∞ onto a strongly logmodular subalgebra of C [X). This is due to a result by K. Hoffman, which states that for every real-valued function u in If, there exists F in (Hx)~1 such that u = log|F|. No notational distinction will be made between an element in If viewed as a function on 3D, or its Gelfand transform viewed as a continuous function on X..