Toeplitz and Hankel Algebras – Axiomatic and Asymptotic Aspects

Abstract

In 1983, the authors introduced a Banach algebra of – as they called them – Toeplitz-like operators. This algebra is defined in an axiomatic way; its elements are distinguished by the existence of four related strong limits. The algebra is in the intersection of Barria and Halmos’ asymptotic Toeplitz operators and of Feintuch’s asymptotic Hankel operators. In the present paper, we start with repeating and extending this approach and introduce Toeplitz and Hankel operators in an abstract and axiomatic manner. In particular, we will see that our abstract Toeplitz operators can be characterized both as shift invariant operators and as compressions. Then we show that the classical Toeplitz and Hankel operators on the spaces ${H}^{p}(\mathbb{T}), {l}^{p}(\mathbb{Z}_{+}), \ \mathrm{and} \ {L}^{p}(\mathbb{R}_{+})$ are concrete realizations of our abstract Toeplitz operators. Finally we generalize some results by Didas on derivations on Toeplitz and Hankel algebras to the axiomatic context.