M ( r , s ) -ideals of compact operators

Rainis Haller; Marje Johanson; Eve Oja

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 3, page 673-693
  • ISSN: 0011-4642

Abstract

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We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces X and Y the subspace of all compact operators 𝒦 ( X , Y ) is an M ( r 1 r 2 , s 1 s 2 ) -ideal in the space of all continuous linear operators ( X , Y ) whenever 𝒦 ( X , X ) and 𝒦 ( Y , Y ) are M ( r 1 , s 1 ) - and M ( r 2 , s 2 ) -ideals in ( X , X ) and ( Y , Y ) , respectively, with r 1 + s 1 / 2 > 1 and r 2 + s 2 / 2 > 1 . We also prove that the M ( r , s ) -ideal 𝒦 ( X , Y ) in ( X , Y ) is separably determined. Among others, our results complete and improve some well-known results on M -ideals.

How to cite

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Haller, Rainis, Johanson, Marje, and Oja, Eve. "$M(r,s)$-ideals of compact operators." Czechoslovak Mathematical Journal 62.3 (2012): 673-693. <http://eudml.org/doc/246277>.

@article{Haller2012,
abstract = {We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces $X$ and $Y$ the subspace of all compact operators $\mathcal \{K\}(X,Y)$ is an $M(r_1 r_2, s_1 s_2)$-ideal in the space of all continuous linear operators $\mathcal \{L\}(X,Y)$ whenever $\mathcal \{K\}(X,X)$ and $\mathcal \{K\}(Y,Y)$ are $M(r_1,s_1)$- and $M(r_2,s_2)$-ideals in $\mathcal \{L\}(X,X)$ and $\mathcal \{L\}(Y,Y)$, respectively, with $r_1+s_1/2>1$ and $r_2+s_2/2>1$. We also prove that the $M(r,s)$-ideal $\mathcal \{K\}(X,Y)$ in $\mathcal \{L\}(X,Y)$ is separably determined. Among others, our results complete and improve some well-known results on $M$-ideals.},
author = {Haller, Rainis, Johanson, Marje, Oja, Eve},
journal = {Czechoslovak Mathematical Journal},
keywords = {$M(r,s)$-ideal and $M$-ideal of compact operators; property $M^\ast (r,s)$; compact approximation property; -ideal; -ideal; property ; compact approximation property},
language = {eng},
number = {3},
pages = {673-693},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$M(r,s)$-ideals of compact operators},
url = {http://eudml.org/doc/246277},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Haller, Rainis
AU - Johanson, Marje
AU - Oja, Eve
TI - $M(r,s)$-ideals of compact operators
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 3
SP - 673
EP - 693
AB - We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces $X$ and $Y$ the subspace of all compact operators $\mathcal {K}(X,Y)$ is an $M(r_1 r_2, s_1 s_2)$-ideal in the space of all continuous linear operators $\mathcal {L}(X,Y)$ whenever $\mathcal {K}(X,X)$ and $\mathcal {K}(Y,Y)$ are $M(r_1,s_1)$- and $M(r_2,s_2)$-ideals in $\mathcal {L}(X,X)$ and $\mathcal {L}(Y,Y)$, respectively, with $r_1+s_1/2>1$ and $r_2+s_2/2>1$. We also prove that the $M(r,s)$-ideal $\mathcal {K}(X,Y)$ in $\mathcal {L}(X,Y)$ is separably determined. Among others, our results complete and improve some well-known results on $M$-ideals.
LA - eng
KW - $M(r,s)$-ideal and $M$-ideal of compact operators; property $M^\ast (r,s)$; compact approximation property; -ideal; -ideal; property ; compact approximation property
UR - http://eudml.org/doc/246277
ER -

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