Abstract Hankel operators in Kreĭn spaces
Let \(\mathcal H_1\), \(\mathcal H_2\) be Krein spaces and let \(T_1\), \(T_2\) be continuous contractions on \(\mathcal H_1\) and \(\mathcal H_2\), respectively, such that the numbers \(\sup \|T_1^n\|\) and \(\sup \|T_1^n\|\) are finite. Then a bounded operator \(X\) from \(\mathcal H_1\) to \(\mathcal H_2\) is called a \textit{Hankel operator} for \(T_1\) and \(T_2\) if \(XT_1^*=T_2X\). The authors give a necessary and sufficient condition for the existence of Hankel symbols for a given Hankel operator \(X\). Here, a \textit{Hankel symbol} \(Z\) is a bounded operator from \(\mathcal K_1\) to \(\mathcal K_2\) such that \(ZU_1^*=U_2Z\) and \(PZ|\mathcal H_1 =X\), where \(U_1:\mathcal K_1\to \mathcal K_1\) and \(U_2: \mathcal K_2\to \mathcal K_2\) are the minimal isometric dilations corresponding to \(T_1\) and \(T_2\), respectively, and \(P\) denotes the orthogonal projection from \(\mathcal K_2\) onto \(\mathcal H_2\). Moreover, a labeling of all Hankel symbols of a Hankel operator \(X\) via a specific set of operator-valued functions analytic on the unit disc and bounded by one (i.e., these functions are from the Schur class) is presented.
scientific article; zbMATH DE number 5774254
scientific article