# Multiplication operators on $L\left({L}_{p}\right)$ and ${\ell}_{p}$-strictly singular operators

William Johnson; Gideon Schechtman

Journal of the European Mathematical Society (2008)

- Volume: 010, Issue: 4, page 1105-1119
- ISSN: 1435-9855

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topJohnson, 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 -

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