Mixed multiplicativity and lp norms for matrices

AbstractLet Cm×n denote the class of m×n complex matrices; and let N1, N2, and N3 be arbitrary norms on Cm×n, Cm×k, and Ck×n, respectively. In this paper we discuss the best (least) positive constant μminwhich satisfies N1(AB)⩽μminN2(A)N3(B) ∀A ∈ Cm×k, B ∈ Ck×n. In particular, for 1 ⩽ p ⩽ ∞ let |A|p= ∑i=1m∑j=1n|αij|p1pbe the lp norm of a matrix A = (αij) ∈ Cm×n. Then for arbitrary p, q, r such that 1⩽ p,q,r ⩽ ∞, we determine explicitly the best constant μmin for which |AB|p ⩽ μmin|A|q|B|r ∀A ∈ Cm×k, B ∀ Ck×n