HOMOTONIC MAPPINGS

Let V be a complex linear space of bounded complex-valued functions de fined on an arbitrary set T. A functional phi: V --> C will be called homotonic if \f\ less than or equal to g implies \phi(f)/ less than or equal to phi(g),f, g is an element of V. The same will hold for a map ping Phi: V --> V from V into itself. In the first part of this paper we obtain bounds for, homotonic functionals, by means of the usual sup norms, parallel to f parallel to(infinity) = suP(t is an element of T ) \f(t)\, f is an element of V. We provide several examples regarding well known functionals on matrices, such as the spectral radius, the n umerical radius, and two families of l(p) norms. The second part of th e paper is devoted to homotonic mappings and to bounds obtained by wei ghted sup norms of the form parallel to f parallel to(w,x) = sup(t is an element of T)\w(t)f(t)\, f is an element of V, where w is a positiv e function, bounded away from zero. Much of the discussion addresses t he case where Vis an associative algebra, and x, the multiplication in V, is homotonic, i.e., \f(1)\ less than or equal to g(1), \f(2)\ less than or equal to g(2) implies \f(1) x f(2)\ 1\ g(1) x g(2) f(1) f(2) g(1) g(2) is an element of V. We give simple conditions on the weight function w that assure power boundedness for parallel to .parallel to( w,x). Our main result proves that if w(-1) is an element of V, then fo r parallel to .parallel to(w,x), multiplicativity, strong stability, a nd quadrativity are each equivalent to the condition w(-2) less than o r equal to w(-1). (C) 1995 Academic Press, Inc.