Submultiplicative functions and operator inequalities

Abstract

Let T : C1(R)→ C(R) be an operator satisfying the “chain rule inequality” T (f ◦ g) ≤ (Tf) ◦ g · Tg ; f, g ∈ C(R) . Imposing a weak continuity and a non-degeneration condition on T , we determine the form of all maps T satisfying this inequality together with T (−Id)(0) 0, H ∈ C(R), A ≥ 1. For A = 1, these are just the solutions of the chain rule operator equation. To prove this, we characterize the submultiplicative, measurable functionsK onR which are continuous in 0 and 1 and satisfy K(−1) 0. Corresponding statements hold in the supermultiplicative case with 0 < A ≤ 1. ∗Supported by Minerva’s Minkowski Center of Tel Aviv University †Supported by Minerva’s Minkowski Center of Tel Aviv University, by the Alexander von Humboldt Foundation, by ISF grant 826/13 and by BSF grant 2012111. 2010 Mathematics Subject Classification: Primary 39B42; Secondary 26A24, 46E15.