On reflexivity and hyperreflexivity of some spaces of intertwining operators

Michal Zajac

Mathematica Bohemica (2008)

  • Volume: 133, Issue: 1, page 75-83
  • ISSN: 0862-7959

Abstract

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Let T , T ' be weak contractions (in the sense of Sz.-Nagy and Foiaş), m , m ' the minimal functions of their C 0 parts and let d be the greatest common inner divisor of m , m ' . It is proved that the space I ( T , T ' ) of all operators intertwining T , T ' is reflexive if and only if the model operator S ( d ) is reflexive. Here S ( d ) means the compression of the unilateral shift onto the space H 2 ⊖ d H 2 . In particular, in finite-dimensional spaces the space I ( T , T ' ) is reflexive if and only if all roots of the greatest common divisor of minimal polynomials of T , T ' are simple. The paper is concluded by an example showing that quasisimilarity does not preserve hyperreflexivity of I ( T , T ' ) .

How to cite

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Zajac, Michal. "On reflexivity and hyperreflexivity of some spaces of intertwining operators." Mathematica Bohemica 133.1 (2008): 75-83. <http://eudml.org/doc/250511>.

@article{Zajac2008,
abstract = {Let $T,T^\{\prime \}$ be weak contractions (in the sense of Sz.-Nagy and Foiaş), $m,m^\{\prime \}$ the minimal functions of their $C_0$ parts and let $d$ be the greatest common inner divisor of $m,m^\{\prime \}$. It is proved that the space $I(T,T^\{\prime \})$ of all operators intertwining $T,T^\{\prime \}$ is reflexive if and only if the model operator $S(d)$ is reflexive. Here $S(d)$ means the compression of the unilateral shift onto the space $H^2\ominus dH^2$. In particular, in finite-dimensional spaces the space $I(T,T^\{\prime \})$ is reflexive if and only if all roots of the greatest common divisor of minimal polynomials of $T,T^\{\prime \}$ are simple. The paper is concluded by an example showing that quasisimilarity does not preserve hyperreflexivity of $I(T,T^\{\prime \})$.},
author = {Zajac, Michal},
journal = {Mathematica Bohemica},
keywords = {intertwining operator; reflexivity; $C_0$ contraction; weak contraction; hyperreflexivity; intertwining operator; reflexivity; contraction; weak contraction},
language = {eng},
number = {1},
pages = {75-83},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On reflexivity and hyperreflexivity of some spaces of intertwining operators},
url = {http://eudml.org/doc/250511},
volume = {133},
year = {2008},
}

TY - JOUR
AU - Zajac, Michal
TI - On reflexivity and hyperreflexivity of some spaces of intertwining operators
JO - Mathematica Bohemica
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 133
IS - 1
SP - 75
EP - 83
AB - Let $T,T^{\prime }$ be weak contractions (in the sense of Sz.-Nagy and Foiaş), $m,m^{\prime }$ the minimal functions of their $C_0$ parts and let $d$ be the greatest common inner divisor of $m,m^{\prime }$. It is proved that the space $I(T,T^{\prime })$ of all operators intertwining $T,T^{\prime }$ is reflexive if and only if the model operator $S(d)$ is reflexive. Here $S(d)$ means the compression of the unilateral shift onto the space $H^2\ominus dH^2$. In particular, in finite-dimensional spaces the space $I(T,T^{\prime })$ is reflexive if and only if all roots of the greatest common divisor of minimal polynomials of $T,T^{\prime }$ are simple. The paper is concluded by an example showing that quasisimilarity does not preserve hyperreflexivity of $I(T,T^{\prime })$.
LA - eng
KW - intertwining operator; reflexivity; $C_0$ contraction; weak contraction; hyperreflexivity; intertwining operator; reflexivity; contraction; weak contraction
UR - http://eudml.org/doc/250511
ER -

References

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