Some inequalities related to Hadamard matrices (Q1812408)

The author defines a parameter \(\rho^{(n)}\) connected with an \(n\times n\) matrix \(A=(a_{ki})\) and a normalized basis \((\varphi_k)\) of a Banach space \(X\) by \[ \rho^{(n)} := \max_{1\leq m\leq 2^n} \Biggl\|\sum_{i=1}^{2^n} \sum_{k=1}^m a_{ki}\varphi_i\Biggr\|. \] Throughout it is assumed that \((\varphi_k)\) is subsymmetric with constant \(1\). The first part deals with the case where \(A\) is a Sylvester matrix. The main result gives an upper estimate for the parameter in terms of the expressions \(\|\sum_{k=1}^n\varphi_k\|\). As corollaries, several other estimates for \(\rho^{(n)}\) and similar parameters are obtained. The value of \(\rho^{(n)}\) for \(l_1\) is given. A general lower estimate is also presented. In a next part, the parameter is evaluated for general Hadamard matrices. Finally an isomorphic characterization of \(l_1\) in terms of the asymptotics of the defined parameters is given. All results are stated without proofs.