Homotonic Mappings

Let V be a complex linear space of bounded complex-valued functions defined on an arbitrary set T. A functional φ: V → C will be called homotonic if |ƒ| ≤ g implies |φ(ƒ)| ≤ φ(g),ƒ, g ∈ V. The same will hold for a mapping Φ: 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, ||ƒ||∞ ≡ supt ∈ T |ƒ(t)|, ƒ ∈ V. We provide several examples regarding well known functionals on matrices, such as the spectral radius, the numerical radius, and two families of lp norms. The second part of the paper is devoted to homotonic mappings and to bounds obtained by weighted sup norms of the form ||ƒ||w,∞ ≡ supt ∈ T |w(t)ƒ(t)|, ƒ ∈ V, where w is a positive function, bounded away from zero. Much of the discussion addresses the case where V is an associative algebra, and ×, the multiplication in V, is homotonic, i.e., |ƒ1| ≤ g1, |ƒ2| ≤ g2 implies |ƒ1 × ƒ2| ≤ g1 × g2, ƒ1, ƒ2, g1, g2 ∈ V. We give simple conditions on the weight function w that assure power boundedness ||•||w,∞. Our main result proves that if w−1 ∈ V, then for ||•||w,∞, multiplicativity, strong stability, and quadrativity are each equivalent to the condition w−2 ≤ w−1.