On the extension of linear contractions: some results of the Hahn-Banach type

Abstract

Let |·| be a norm on R n , and s a compact convex semigroup of linear |·|-contractions. Given two k -tuples of n -vectors, ( x (1) ,…, x ( k ) ) and ( y (1) ,…, y ( k ) ), we seek conditions for the existence of a contraction S ϵ s that simultaneously takes x ( i ) to y ( i ) , that is, Sx ( i ) = y ( i ) , for all i = 1,…, k !. Straightforward application of the separation theorem for convex sets provides a general but abstract result, in the form of a system of inequalities. Specializing |·| to the 1-norm and the ∞-norm, respectively, and s to comprise either all contractions or those contractions that preserve a particular linear form, it is possible to evaluate the characteristic functionals arising from the separation theoem. Therby the abstract results can be rduced to a tractable form, which turns out to be of the Hahn-Banach type.