Toeplitz and Hankel operators with aHandy formula and Localization Principle

Abstract

We give a decomposition theorem for the Sobolev space of first order on the disc. Using this result, some characterizations for algebraic properties of Toeplitz or small Hankel operators with symbols in 𝐿∞,1 are given. We consider Toeplitz and Hankel operators with piecewise continuous generating functions on 𝑙𝑝-spaces and the Banach algabra generated by them. We characterize the pairs of truncated Hankel operators on the model spaces, the asymptotic behavior of the singular values of a compact Hankel operator is determined by the behavior of the symbol in a neighbourhood of its singular support.