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Duality and Hankel operators on the Bergman spaces of bounded symmetric domains. (English) Zbl 0669.47019
L\({}^ p_ a(\Omega)\) denotes the Bergman space in a bounded symmetric domain \(\Omega\) in \({\mathbb{C}}^ n\). The author describes the dual and predual of \(L^ 1_ a(\Omega)\) in terms of the (reduced) Hankel operators and the compact ones, respectively. Let \({\mathcal B}(\Omega)\) be the Bloch space of \(\Omega\) and \({\mathcal B}_ 0(\Omega)\) be the little Bloch space of \(\Omega\). He shows the following: \({\mathcal B}(\Omega)\subseteq the\) dual of \(L^ 1_ a(\Omega)\) and \({\mathcal B}_ 0(\Omega)\subseteq the\) predual of \(L^ 1_ a(\Omega)\). Moreover, equalities hold iff rank \(\Omega\) \(=1\). For the polydisc \(D^ n\) \((n>1)\), he describes the symbols of the bounded Hankel operators and the compact ones.
Reviewer: T.Nakazi

MSC:
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
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