×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI Link