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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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##### References:
 [1] \scJ. Arazy, S. D. Fisher, and J. Peetre, Hankel operators on weighted Bergman spaces, to appear. Amer. J. Math. [2] Axler, S., Bergman spaces and their operators, () · Zbl 0681.47006 [3] Berger, C.A.; Coburn, L.A.; Zhu, K.H., BMO on the Bergman spaces of bounded symmetric domains, Bull. amer. math. soc., 17, 133-136, (1987) · Zbl 0621.32014 [4] Coifman, R.R.; Rochberg, R., Representation theorems for holomorphic and harmonic functions in Lp, Astérisque, 77, 11-66, (1980) · Zbl 0472.46040 [5] Chang, S.Y.A.; Fefferman, R., A continuous version of duality of H1 with BMO on the bidisc, Ann. of math., 112, 179-201, (1980) · Zbl 0451.42014 [6] Fefferman, R., Bounded Mean oscillation on the polydisc, Ann. of math., 110, 395-406, (1979) · Zbl 0429.32016 [7] Forelli, F.; Rudin, W., Projections on spaces of holomorphic functions in balls, Indiana univ. math. J., 24, 593-602, (1974) · Zbl 0297.47041 [8] Stoll, M., Mean value theorems for harmonic and holomorphic functions on bounded symmetric domains, J. reine. angew. math., 283, 191-198, (1977) · Zbl 0342.32003 [9] Timoney, R.; Timoney, R., Bloch functions in several complex variables, II, Bull. London math. soc., J. reine angew. math., 319, 1-22, (1980) · Zbl 0425.32008
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