# Multipliers and functional analysis

1 AF - Analyse fonctionnelle
IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
Abstract : We study several functional properties of unconditionality and state them as a property of families of multipliers. This Thesis has three parts. Part I is devoted to the study of several notions of isometric and almost isometric unconditionality in separable Banach spaces. The most general such notion is that of metric unconditional approximation property''. We characterize this umap'' by a simple property of block unconditionality'' for spaces with nontrivial cotype. We focus on subspaces of Banach spaces of functions on the circle spanned by a sequence of characters $e^(int)$. There umap may be stated in terms of Fourier multipliers. We express umap as a simple combinatorial property of this sequence. We obtain a corresponding result for isometric and almost isometric basic sequences of characters. Our study uses the following crucial property of the $L^p$ norm for even $p$: $\int |f|^p = \int (|f^(p/2)|)^2 = \sum |\widehat(f^(p/2))(n)|^2$ is a polynomial expression in the Fourier coefficients of $f$ and $\bar f$. As a byproduct, we get a sharp estimate of the Sidon constant of sets à la Hadamard. Part II studies Schur multipliers: we characterize isometric unconditional basic sequences of matrix entries $e_(ij)$ in the Schatten class $S^p$. The combinatorial properties that we obtain concern paths on the lattice $\N\times\N$ with vertices in this set. Part III studies the relationship between the growth rate of an integer sequence and harmonic and functional properties of the corresponding sequence of characters. We show in particular that every polynomial sequence contains a set that is $\Lambda(p)$ for all $p$ but is not a Rosenthal set. This holds also for the sequence of primes.
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Cited literature [66 references]

https://tel.archives-ouvertes.fr/tel-00010399
Contributor : Stefan Neuwirth <>
Submitted on : Wednesday, October 5, 2005 - 11:40:18 AM
Last modification on : Monday, January 4, 2021 - 9:58:03 PM
Long-term archiving on: : Friday, April 2, 2010 - 10:34:54 PM

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• HAL Id : tel-00010399, version 1

### Citation

Stefan Neuwirth. Multipliers and functional analysis. Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 1999. English. ⟨tel-00010399⟩

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