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Boundedness and compactness of generalized Hankel operators on bounded symmetric domains. (English) Zbl 0880.47015
Let $$D$$ be a domain in $$\mathbb{C}^n$$, the Hankel operator associated with an analytic function $$b$$ on $$D$$ is $$H_b=(I-P)M_{\overline b}P$$, where $$M_{\overline b}$$ is the multiplication by $$\overline b$$, and $$P$$ is a projection.
In this paper, when $$D$$ is a Cartan domain of tube type, of dimension $$n$$, and of rank $$r>1$$, in $$\mathbb{C}^n$$, the author introduces the generalization $$A_b$$ of the Hankel operator $$H_b$$, the generalized Bloch and little Bloch spaces $$B$$, $$B_0$$, and the generalized invariant oscillation spaces BMOA, VMOA.
Main results of the author:
$$A_b$$ is bounded if and only if $$b\in B$$ if and only if $$b\in\text{BMOA}$$.
$$A_b$$ is compact if and only if $$b\in B_0$$ if and only if $$b\in\text{VMOA}$$.
$$A_b$$ belongs to the Hilbert-Schmidt class $$S_2$$ if and only if $$b$$ is in the generalized Besov space $$B_2$$.
Reviewer: B.D.Khanh (Paris)

##### MSC:
 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.) 47B07 Linear operators defined by compactness properties
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