Norms of Schur multipliers (Q5917602)

This note deals with the characterization of the extremal \(m\times n\) matrices \(A\) for which the inequality due to \textit{G. Bennett} [Duke Math. J. 44, 603-639 (1977; Zbl 0389.47015)], \(\| A\|_{(p, q)}\leq \| A\|_{p, q}\) turns into an equality, where \(\| A\|_{p, q}= \sup\{\| Ax\|_ q: \| x\|_ p\leq 1\}\), \(\| A\|_{(p, q)}= \sup\{\| A\cdot B\|_{p, q}: \| B\|_{p, q}\leq 1\}\), and \(A\cdot B\) denotes the Schur product (the Hadamard product) of two \(n\times n\) complex matrices: \((a_{ij} b_{ij})\), \(1\leq p\), \(q\leq \infty\). The main result is Theorem 1. Let \(A\) be an \(m\times n\) complex matrix, and \(1\leq p\leq \infty\), \(p\neq 2\). The following conditions are equivalent: (a) \(\| A\|_{p, p}= \| A\|_{(p, p)}\), (b) \(\| A\cdot \overline A\|_{p, p}= \| A\|^ 2_{p, p}\), (c) there exist a scalar \(\lambda\geq 0\), two permutation matrices \(P\) and \(Q\) of sizes \(m\times m\) and \(n\times n\) respectively, \(k\) complex numbers \(\varepsilon_ 1,\dots, \varepsilon_ k\) with \(|\varepsilon_ 1|=\cdots= |\varepsilon_ k|= 1\), \(1\leq k\leq m\), and an \((m- k)\times (n- k)\) \((p, q)\)-contraction \(C\) such that \[ PAQ= \lambda\left[\begin{matrix} \varepsilon_ 1 && 0\\ & \ddots && 0\\ 0 && \varepsilon_ k\\ & 0 && C\end{matrix}\right]. \] The above inequality is an extension of the classical result of Schur, namely, the case when \(p= q= 2\). The author's result characterizes \(m\times n\) matrices with extremal \((p, q)\)-Schur multiplier norms.