HADAMARD FUNCTIONS PRESERVING NONNEGATIVE H-MATRICES

For k nonnegative n x n matrices A(l) = (a(ij)(l)) and a function f : R<INF>+</INF><SUP>k</SUP> --> R<INF>+ </INF>consider the matrix C = f( A(l),...,A(k)) = (C-ij), where c(ij) = f(a(ij)(l),...,a(ij)(k)), i,j=l ,...,n. Denote by rho(A) the spectral radius of a nonnegative square m atrix A, and by sigma(A) the minimal real eigenvalue of its comparison matrix M(A) = 2 diag(a(ii)) -A. It is known that the function f(x(1), ...,x(k)) = cx(1)(alpha 1)...x(k)(alpha k), where alpha(i) is an eleme nt of R+, Sigma(i=l)(k) alpha(i) greater than or equal to 1 and c > 0, satisfies the inequalities rho(f(A(1),...,A(k))) less than or equal t o f(rho(A(1)),...,rho(A(k))), as well as the in equalities sigma(f(A(1 ),...,A(k))) greater than or equal to f(sigma(A(1)),...,sigma(A(k))). whenever A(i) are nonnegative H-matrices, i.e. sigma(A(i)) greater tha n or equal to 0. The last inequality implies that the above function f ` maps the set of nonnegative I-I-matrices into itself. In this note i t is proven that these are the only continuous functions with this pro perty. (C) 1998 Elsevier Science Inc. AU rights reserved.