Tensor products of functors on categories of Banach spaces

Abstract

In their fundamental paper [11~ B.S. MITYAGIN and A.S. SHVARTS have laid the foundations for a theory of functors on categories of Banach spaces. The situation may be roughly described as follows: The family Ban of all Banach spaces becomes a category by choosing as morphisms al~ linear contractions, i.e. all bounded linear mappings ~: X * Y satisfying II~II% I. The set of all morphisms from X into Y may therefore be identified with the unit ball of the Banach space H(X,Y) of all bounded linear maps from X into Y. By a (covariant) functor F: Ban * Ban we mean a functor in the algebraic sense with the additional property that the mapping f * F(f) is a linear contraction from H(X,Y) into H(F(X), F(Y)) for all X,Y. The simplest examples are the functors E A and H A defined by EA(X)=A ~ X (i.e. the projective tensor product) and HA(X)=H(A,X).