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.