Geodesic
On Pták’s generalization of Hankel operators
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 2, pp. 323-342 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In 1997 Pták 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 $X \:{\mathcal H}_1 \rightarrow {\mathcal H}_2$ is said to be a generalized Hankel operator if $T_2X=XT_1^*$ 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 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 $T_1$ 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 Pták.
In 1997 Pták 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 $X \:{\mathcal H}_1 \rightarrow {\mathcal H}_2$ is said to be a generalized Hankel operator if $T_2X=XT_1^*$ 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 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 $T_1$ 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 Pták.
Classification : 46E22, 47A20, 47B35
Keywords: Toeplitz operators; Hankel operators; minimal isometric dilation 
@article{CMJ_2001_51_2_a7,
     author = {Mancera, Carmen H. and Pa\'ul, Pedro J.},
     title = {On {Pt\'ak{\textquoteright}s} generalization of {Hankel} operators},
     journal = {Czechoslovak Mathematical Journal},
     pages = {323--342},
     year = {2001},
     volume = {51},
     number = {2},
     mrnumber = {1844313},
     zbl = {0983.47019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2001_51_2_a7/}
}
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Mancera, Carmen H.; Paúl, Pedro J. On Pták’s generalization of Hankel operators. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 2, pp. 323-342. http://geodesic.mathdoc.fr/item/CMJ_2001_51_2_a7/