On Pták's generalization of Hankel operators

Abstract

In 1997 Pták defined generalized Hankel operators as follows: Given two contractions T1 ∈ B(H1) and T2 ∈ B(H2), an operator X: H1 → H2 is said to be a generalized Hankel operator if T2X = XT ∗ 1 and X satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations of T1 and T2. This approach, call it (P), contrasts with a previous one developed by Pták and Vrbová 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 T1 and T2, 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 Pták.