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In 1997 Pták defined generalized Hankel operators as follows: Given two contractions $T_1\in ℬ\left({\mathscr{H}}_1\right)$ and $T_2\in ℬ\left({\mathscr{H}}_2\right)$, an operator $X{\mathscr{H}}_1\to {\mathscr{H}}_2$ is said to be a generalized Hankel operator if $T_{2}X=X{T}_1^{*}$ 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 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...