Multiplication operators on L(Lp) and lp-strictly singular operators Academic Article

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 l p-strictly singular operators, and we also investigate the structure of general lp-strictly singular operators on Lp. The main result is that if an operator T on Lp, 1 < p < 2, is l 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 Lp 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. © European Mathematical Society 2008.

authors

• Johnson, William

author list (cited authors)

• Johnson, W., & Schechtman, G.

complete list of authors

• Johnson, William||Schechtman, Gideon

publication date

• January 1, 2008 11:11 AM

publisher

• European Mathematical Society - EMS - Publishing House GmbH  Publisher
• European Mathematical Society Publishing House  Publisher

published in

• Journal of the European Mathematical Society  Journal

keywords

• Biased Coin
• Elementary Operators
• L-p Spaces
• Multiplication Operators
• Strictly Singular Operators

Digital Object Identifier (DOI)

• 10.4171/jems/141

start page

• 1105

end page

• 1119

volume

• 10

issue

• 4