Approximate Identities, Almost-Periodic Functions and Toeplitz Operators

A sequence {A } of linear bounded operators is called stable if, for all sufficiently large , the inverses of A exist and their norms are uniformly bounded. We consider the stability problem for sequences of Toeplitz operators {T(k a)}, where a(t) is an almost-periodic function on unit circle and k a is an approximate identity. A stability criterion is established in terms of the invertibility of a family of almost-periodic functions. This family of functions depends on the approximate identity used in a very subtle way, and the stability condition is, in general, stronger than the invertibility condition of the Toeplitz operator T(a).