A remark on the range of elementary operators

Said Bouali; Youssef Bouhafsi

Czechoslovak Mathematical Journal (2010)

  • Volume: 60, Issue: 4, page 1065-1074
  • ISSN: 0011-4642

Abstract

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Let L ( H ) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Given A ∈ L ( H ) , we define the elementary operator Δ A : L ( H ) ⟶ L ( H ) by Δ A ( X ) = A X A - X . In this paper we study the class of operators A ∈ L ( H ) which have the following property: A T A = T implies A T * A = T * for all trace class operators T ∈ C 1 ( H ) . Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of Δ A is closed under taking adjoints. We give a characterization and some basic results concerning generalized quasi-adjoints operators.

How to cite

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Bouali, Said, and Bouhafsi, Youssef. "A remark on the range of elementary operators." Czechoslovak Mathematical Journal 60.4 (2010): 1065-1074. <http://eudml.org/doc/197033>.

@article{Bouali2010,
abstract = {Let $L(H)$ denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space $H$ into itself. Given $A\in L(H)$, we define the elementary operator $\Delta _A\colon L(H)\longrightarrow L(H)$ by $\Delta _A(X)=AXA-X$. In this paper we study the class of operators $A\in L(H)$ which have the following property: $ATA=T$ implies $AT^\{\ast \}A=T^\{\ast \}$ for all trace class operators $T\in C_1(H)$. Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of $\Delta _A$ is closed under taking adjoints. We give a characterization and some basic results concerning generalized quasi-adjoints operators.},
author = {Bouali, Said, Bouhafsi, Youssef},
journal = {Czechoslovak Mathematical Journal},
keywords = {elementary operators; ultraweak closure; weak closure; quasi-adjoint operator; elementary operator; ultraweak closure; weak closure; generalised quasi-adjoint operator},
language = {eng},
number = {4},
pages = {1065-1074},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A remark on the range of elementary operators},
url = {http://eudml.org/doc/197033},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Bouali, Said
AU - Bouhafsi, Youssef
TI - A remark on the range of elementary operators
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 4
SP - 1065
EP - 1074
AB - Let $L(H)$ denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space $H$ into itself. Given $A\in L(H)$, we define the elementary operator $\Delta _A\colon L(H)\longrightarrow L(H)$ by $\Delta _A(X)=AXA-X$. In this paper we study the class of operators $A\in L(H)$ which have the following property: $ATA=T$ implies $AT^{\ast }A=T^{\ast }$ for all trace class operators $T\in C_1(H)$. Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of $\Delta _A$ is closed under taking adjoints. We give a characterization and some basic results concerning generalized quasi-adjoints operators.
LA - eng
KW - elementary operators; ultraweak closure; weak closure; quasi-adjoint operator; elementary operator; ultraweak closure; weak closure; generalised quasi-adjoint operator
UR - http://eudml.org/doc/197033
ER -

References

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