On the Coefficient Problem: a Version of the Kahane–Katznelson–De Leeuw Theorem for Spaces of Matrices
✍ Scribed by Françoise Lust-Piquard
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 430 KB
- Volume
- 149
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
It is known that, for every (a n ) # l 2 (Z) there exists a function F # C(T) such that |a n | |F (n)| for every n # Z. We prove a noncommutative version: for every matrix A=(a ij ) such that sup i &(a ij ) j & l 2 and sup j &(a ij ) i & l 2 are finite, there exists a matrix (b ij ) defining a bounded operator on l 2 , such that |a ij | |b ij | for every i, j. We extend this to other norms on matrices and present an abstract version of the coefficient problem.
1997 Academic Press Kahane, Katznelson, de Leeuw proved the following:
Theorem 1 (KKL). For every sequence a=(a j ) j # Z # l 2 (Z) there exists a function F # C(T ) such that
where K is an absolute constant.
Recently Nazarov [N] gave several simple proofs of a stronger result (see Theorem 5 below). We show how his methods may be used in a noncommutative setting.
We prove:
Theorem 2. For every matrix A=(a ij ) 0 i, j such that A and A* # l (l 2 ), there exists a matrix B=(b ij ) 0 i, j defining a bounded operator: l 2 Ä l 2 such that
where K is an absolute constant.