Multiplication operators on L ( L p ) and ℓ 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

Abstract

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A classification of weakly compact multiplication operators on L ( L p ) , 1<p< , i s g i v e n . T h i s a n s w e r s a q u e s t i o n r a i s e d b y S a k s m a n a n d T y l l i i n 1992 . T h e c l a s s i f i c a t i o n i n v o l v e s t h e c o n c e p t o f p - s t r i c t l y s i n g u l a r o p e r a t o r s , a n d w e a l s o i n v e s t i g a t e t h e s t r u c t u r e o f g e n e r a l p - s t r i c t l y s i n g u l a r o p e r a t o r s o n Lp . T h e m a i n r e s u l t i s t h a t i f a n o p e r a t o r T o n Lp , 1<p<2 , i s p - s t r i c t l y s i n g u l a r a n d T|X i s a n i s o m o r p h i s m f o r s o m e s u b s p a c e X o f Lp , t h e n X e m b e d s i n t o Lr f o r a l l r<2 , b u t X n e e d n o t b e i s o m o r p h i c t o a H i l b e r t s p a c e . It is also shown that if T is convolution by a biased coin on L p of the Cantor group, 1 ≤ 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.

How to cite

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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 -

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