Type norms with respect to characters of compact abelian groups and algebraic relations

Let A _n =(a ₁,...,a _n) be a system of characters of a compact abelian group A _n with normalized Haar measure μ and let T be a bounded linear operator from a Banach space X into a Banach space Y. The type norm τ(T| A _n ) of T with respect to A _n is the least constant c such that for all x ₁,..., x _n ∈ X. We investigate under which conditions on two systems A _n and ℬ _n of characters of compact abelian groups an inequality τ(T|ℬ _n) ≦ τ(T|A _n ) holds for all linear bounded operators T between Banach spaces. It turns out that this can be tested on a certain operator depending only on the system ℬ _n. Moreover, it is equivalent to strong algebraic relations between A _n and ℬ _n as well as to relations between its distributions. In particular, for systems of trigonometric functions this inequality for all linear bounded operators even implies equality for all linear bounded operators. The author is supported by DFG grant PI 322/1-1. The content of this paper is part of the authors PhD-thesis written under the supervision of A. Pietsch.