Boundedness and compactness of generalized Hankel operators on bounded symmetric domains
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 \(B_2\).
generalized Bloch and little Bloch spaces
generalized invariant oscillation spaces
scientific article; zbMATH DE number 871731