Multiplicativity and compatibility of generalized matrix norms (Q1239219)

lt is well known that if a generalized matrix norm (gmn) is multiplicative, then it has a compatible vector norm associated with it. The converse, however, is invalid, and the precise relation between multiplicativity and compatibility is explored for a gmn. Theorem 3: Every vector norm is compatible with a non- multiplicative gmn. Theorem 4: Suppose \(G\) is a gmn on \(Mk (c)\), and let \(\rho\) denote the spectral radius. There is a vector norm compatible with \(G\) if and only if \(G\) satisfies \[ \prod_{i=1}^m G(A_i) \geq \rho(A_1A_2\dots A_m) \] for all integers \(m\) and all sequences \(A_1,\dots,A_m\); \(A_i\in M_k(C)\), \(i=1,\dots,m\). Also, certain methods for deriving one norm from another are mentioned (one of which is based on the Hadamard product), and several questions of interest in the area are raised.