On the Coefficient Problem: a Version of the Kahane–Katznelson–De Leeuw Theorem for Spaces of Matrices

It is known that, for every (an)∈l2(Z) there exists a functionF∈C(T) such that |an|⩽|F(n)| for everyn∈Z. We prove a noncommutative version: for every matrixA=(aij) such that supi Verbar;(aij)j‖l2and supj ‖(aij)i‖l2are finite, there exists a matrix (bij) defining a bounded operator onl2, such that |aij|⩽|bij| for everyi, j. We extend this to other norms on matrices and present an abstract version of the coefficient problem.