Hankel operators on bounded analytic functions. (English) Zbl 0942.47019
\(H^\infty(U)\) denotes the algebra of bounded analytic functions on a bounded open subset \(U\) of the complex plane. \(H^\infty(U)\) is regarded as a subalgebra of \(L^\infty(\sigma)\) where \(\sigma\) denotes area measure on \(U\). For fixed \(g\in L^\infty(\sigma)\), \(S_g(f)= gf+ H^\infty(U)\) and \(S_g\) is called Hankel-type operator from \(H^\infty(U)\) to \(L^\infty(\sigma)/H^\infty(U)\). Put \(B^\infty(U)= \{g\in L^\infty(\sigma): S_g\) is compact}. In general, \(B^\infty(U)\supseteq [H^\infty(U)+ C(\overline U)+ L^\infty_0(\sigma)]\) where \([X]\) denotes the closed linear span in \(L^\infty(\sigma)\), \(C(\overline U)\) is the set of all continuous functions on \(\overline U\) and \(L^\infty_0(\sigma)\) is the set of those functions from \(L^\infty(\sigma)\) that tend to \(0\) at \(\partial U\).
The authors show that if every point of \(\partial U\) is a peak point for \(H^\infty(U)\), then \(B^\infty(U)= [H^\infty(U)+ C(\overline U)+ L^\infty_0(\sigma)]\). Moreover, they show that \(B^\infty(U)\neq [H^\infty(U)+ C(\overline U)+ L^\infty_0(\sigma)]\) for the class of infinitely connected domains introduced by M. Behrens.
The authors show that if every point of \(\partial U\) is a peak point for \(H^\infty(U)\), then \(B^\infty(U)= [H^\infty(U)+ C(\overline U)+ L^\infty_0(\sigma)]\). Moreover, they show that \(B^\infty(U)\neq [H^\infty(U)+ C(\overline U)+ L^\infty_0(\sigma)]\) for the class of infinitely connected domains introduced by M. Behrens.
Reviewer: T.Nakazi (Sapporo)
MSC:
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 46J15 | Banach algebras of differentiable or analytic functions, \(H^p\)-spaces |
| 47B38 | Linear operators on function spaces (general) |
| 46E10 | Topological linear spaces of continuous, differentiable or analytic functions |
| 30D55 | \(H^p\)-classes (MSC2000) |
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