Multiplication operators on $L\left(L_p\right)$ and ${\ell }_p$-strictly singular operators

Johnson, William, and Schechtman, Gideon. "Multiplication operators on $L(L_p)$ and $\ell _p$-strictly singular operators." Journal of the European Mathematical Society 010.4 (2008): 1105-1119. <http://eudml.org/doc/277694>.

@article{Johnson2008,
abstract = {A classification of weakly compact multiplication operators on $L(L_p), $1<p<$, is given. This answers a question raised by Saksman and Tylli in 1992. The classification involves the concept of $p$-strictly singular operators, and we also investigate the structure of general $p$-strictly singular operators on $Lp$. The main result is that if an operator $T$ on $Lp$, $1<p<2$, is $p$-strictly singular and $T|X$ is an isomorphism for some subspace $X$ of $Lp$, then $X$ embeds into $Lr$ for all $r<2$, but $X$ need not be isomorphic to a Hilbert space. $It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p<2$, and $T_\{|X\}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.},
author = {Johnson, William, Schechtman, Gideon},
journal = {Journal of the European Mathematical Society},
keywords = {elementary operators; multiplication operators; strictly singular operators; $L_p$ spaces; biased coin; elementary operator; multiplication operators; strictly singular operators; spaces; biased coin},
language = {eng},
number = {4},
pages = {1105-1119},
publisher = {European Mathematical Society Publishing House},
title = {Multiplication operators on $L(L_p)$ and $\ell _p$-strictly singular operators},
url = {http://eudml.org/doc/277694},
volume = {010},
year = {2008},
}

TY - JOUR
AU - Johnson, William
AU - Schechtman, Gideon
TI - Multiplication operators on $L(L_p)$ and $\ell _p$-strictly singular operators
JO - Journal of the European Mathematical Society
PY - 2008
PB - European Mathematical Society Publishing House
VL - 010
IS - 4
SP - 1105
EP - 1119
AB - A classification of weakly compact multiplication operators on $L(L_p), $1<p<$, is given. This answers a question raised by Saksman and Tylli in 1992. The classification involves the concept of $p$-strictly singular operators, and we also investigate the structure of general $p$-strictly singular operators on $Lp$. The main result is that if an operator $T$ on $Lp$, $1<p<2$, is $p$-strictly singular and $T|X$ is an isomorphism for some subspace $X$ of $Lp$, then $X$ embeds into $Lr$ for all $r<2$, but $X$ need not be isomorphic to a Hilbert space. $It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p<2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.
LA - eng
KW - elementary operators; multiplication operators; strictly singular operators; $L_p$ spaces; biased coin; elementary operator; multiplication operators; strictly singular operators; spaces; biased coin
UR - http://eudml.org/doc/277694
ER -