Norms of Hankel operators and uniform algebras. II (Q1090905)

Let \(H^{\infty}\) be an abstract Hardy space associated with a uniform algebra. Denoting by (f) the coset in \((L^{\infty})^{- 1}/(H^{\infty})^{-1}\) of an f in \((L^{\infty})^{-1}\), define \(\| (f)\| =\inf \{\| g\| _{\infty}\| g^{-1}\| _{\infty}\); \(g\in (f)\}\) and \(\gamma _ 0=\sup \{\| (f)\|\); \((f)\in (L^{\infty})^{-1}/(H^{\infty})^{-1}\}\). If \(\gamma _ 0\) is finite, the author shows that the norms of Hankel operators are equivalent to the dual norms of \(H^ 1\) or the distances of the symbols of Hankel operators. For the proof, he uses his previous result of part I [Trans. Am. Math. Soc. 299, 573-580 (1987)]. If \(H^{\infty}\) is the algebra of bounded analytic functions on a multiply connected domain, then he shows that \(\gamma _ 0\) is finite and he determines the essential norms of Hankel operators. \(\gamma _ 0=1\) for the classical \(H^{\infty}\) on the unit disc and \(\gamma _ 0=1/\sqrt{r}\) for the annulus algebra on \(\{\) \(z: r<| z| <1\}\).