On the Structure of Hankel Algebras

Abstract

Let H 2 denote the Hardy space on the unit disc \({\mathbb{D}}\) and let A be a closed subalgebra of \({L^\infty(\partial\mathbb{D})}\) strictly containing H ∞. The Hankel algebra \({\fancyscript{H}_A}\) is the smallest closed subalgebra of B(H 2) containing all Toeplitz and Hankel operators with symbols from A. We establish a short exact sequence of the form \({0 \to \fancyscript{C} \to \fancyscript{H}_A \to A \to 0}\) generalizing the corresponding sequence for the underlying Toeplitz algebra, where \({\fancyscript{C}}\) denotes the commutator ideal of \({\fancyscript{H}_A}\). This extends a result of Power (J Funct Anal 31:52–68, 1979) to the non-selfadjoint setting. By a similar method we obtain a decomposition theorem for the set of all operators \({X \in B(H^2)}\) that are simultaneously asymptotically Toeplitz and Hankel (in the sense of Barria-Halmos, Trans Am Math Soc 273(2):621–630, 1982) and Feintuch, J Funct Anal 94:1–13, 1990, respectively). As an application of the above short exact sequence we show that every derivation on \({\fancyscript{H}_A}\) is a commutator with an operator \({S \in B(H^2)}\) and maps into the commutator ideal.