Strategically reproducible bases and the factorization property
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STRATEGICALLY REPRODUCIBLE BASES AND THE FACTORIZATION PROPERTY BY
Richard Lechner∗ Institute of Analysis, Johannes Kepler University Linz Altenberger Strasse 69, A-4040 Linz, Austria e-mail: [email protected] AND
Pavlos Motakis∗∗ Department of Mathematics, University of Illinois at Urbana-Champaign Urbana, IL 61801, USA e-mail: [email protected] AND
¨ller∗ Paul F. X. Mu Institute of Analysis, Johannes Kepler University Linz Altenberger Strasse 69, A-4040 Linz, Austria e-mail: [email protected] AND
Thomas Schlumprecht† Department of Mathematics, Texas A&M University College Station, TX 77843-3368, USA and Faculty of Electrical Engineering, Czech Technical University in Prague Technika 2, 16627, Praha 6, Czech Republic e-mail: [email protected]
∗ The first and the third author were supported by the Austrian Science Foundation
(FWF) under Grant Number Pr. Nr. P28352.
∗∗ The second-named author was supported by the National Science Foundation
under Grant Numbers DMS-1600600 and DMS-1912897. † The fourth-named author was supported by the National Science Foundation un-
der Grant Numbers DMS-1464713 and DMS-1711076. Received September 20, 2018 and in revised form December 18, 2018
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R. LECHNER ET AL.
Isr. J. Math.
ABSTRACT
We introduce the concept of strategically reproducible bases in Banach spaces and show that operators which have large diagonal with respect to strategically reproducible bases are factors of the identity. We give several examples of classical Banach spaces in which the Haar system is strategically reproducible: multi-parameter Lebesgue spaces, mixed-norm Hardy spaces and most significantly the space L1 . Moreover, we show the strategical reproducibility is inherited by unconditional sums.
1. Introduction In this paper, we address the following question: Given a Banach space X with a basis (ei )∞ i=1 , let T : X → X be an operator, whose matrix representation has a diagonal whose elements are uniformly bounded away from 0. We say in that case that T has a large diagonal. Is it possible to factor the identity operator on X through T ? The origin of this problem can be traced back to the work of Pelczy´ nski [24], who proved that every infinite-dimensional subspace of ℓp , 1 ≤ p < ∞ and c0 contains a further subspace which is complemented and isomorphic to the whole space. Closely related is the concept of primarity of a Banach space. Recall that X is called primary, if for every bounded projection P : X → X, either P (X) or (I − P )(X) is isomorphic to X. The connection between the primarity of a Banach space and the factorization problem is as follows: either P has large diagonal or I −P has large diagonal on a “large” subsequence of the basis (ei )∞ i=1 of the Banach space X. For example Enflo (according to [17]) proved primarity for X = Lp , 1 ≤ p < ∞, by showing that for every operator T : Lp → Lp , the identity operator factors either through T or I − T ; see also Alspach–Enflo– Odell [1]. Factorization and primarity theorems were obtained by Capon [4] for the mixed nor
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