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 ab elian group G with normalized Haar measure mu and let T be a bounded l inear operator from a Banach space X into a Banach space Y. The type n orm tau(T\A(n)) of T with respect to A(n) is the least constant c such that [GRAPHICS] for all x(1), ..., x(n) is an element of X. We invest igate under which conditions on two systems A(n) and B-n of characters of compact abelian groups an inequality tau(T\B-n) less than or equal to tau(T\A(n)) holds for all linear bounded operators T between Banac h spaces. It turns out that this can be tested on a certain operator d epending 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 betwe en its distributions. In particular, for systems of trigonometric func tions this inequality for all linear bounded operators even implies eq uality for all linear bounded operators.