On Ptak's Generalization of Hankel Operators

Abstract

In 1997 Ptak defined generalized Hankel operators as follows: Given two contractions \(T_1 \in \mathcal{B}(\mathcal{H}_1 )\) and \(T_2 \in \mathcal{B}(\mathcal{H}_2 )\), an operator \(X:\mathcal{H}_1 \to \mathcal{H}_2 \) is said to be a generalized Hankel operator if \(T_2 X = XT_1^* \) and X satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations of T 1 and T 2. This approach, call it (P), contrasts with a previous one developed by Ptak and Vrbova in 1988, call it (PV), based on the existence of a previously defined generalized Toeplitz operator. There seemed to be a strong but somewhat hidden connection between the theories (P) and (PV) and we clarify that connection by proving that (P) is more general than (PV), even strictly more general for some T 1 and T 2, and by studying when they coincide. Then we characterize the existence of Hankel operators, Hankel symbols and analytic Hankel symbols, solving in this way some open problems proposed by Ptak.