On Pták’s generalization of Hankel operators

Carmen H. Mancera; Pedro José Paúl

Czechoslovak Mathematical Journal (2001)

  • Volume: 51, Issue: 2, page 323-342
  • ISSN: 0011-4642

Abstract

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In 1997 Pták defined generalized Hankel operators as follows: Given two contractions T 1 ∈ ℬ ( ℋ 1 ) and T 2 ∈ ℬ ( ℋ 2 ) , an operator X ℋ 1 → 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 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.

How to cite

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Mancera, Carmen H., and Paúl, Pedro José. "On Pták’s generalization of Hankel operators." Czechoslovak Mathematical Journal 51.2 (2001): 323-342. <http://eudml.org/doc/30637>.

@article{Mancera2001,
abstract = {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.},
author = {Mancera, Carmen H., Paúl, Pedro José},
journal = {Czechoslovak Mathematical Journal},
keywords = {Toeplitz operators; Hankel operators; minimal isometric dilation; Toeplitz operators; Hankel operators; minimal isometric dilation; analytic Hankel symbols},
language = {eng},
number = {2},
pages = {323-342},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On Pták’s generalization of Hankel operators},
url = {http://eudml.org/doc/30637},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Mancera, Carmen H.
AU - Paúl, Pedro José
TI - On Pták’s generalization of Hankel operators
JO - Czechoslovak Mathematical Journal
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 2
SP - 323
EP - 342
AB - 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.
LA - eng
KW - Toeplitz operators; Hankel operators; minimal isometric dilation; Toeplitz operators; Hankel operators; minimal isometric dilation; analytic Hankel symbols
UR - http://eudml.org/doc/30637
ER -

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