The Common Causality Structure of Multilinear Maps and Their Multipower Forms*

This paper deals with the causality structure of multilinear, multipower operators on a Hilbert resolution space. One objective of this study is to clear up certain unanswered questions raised in [l]. In particular we present a counterexample consisting of a compact set K C&(0, 1) such that no memoryless multipower operators of order n >, 2 can be defined on K. This lays to rest the tempting conjecture that a Weierstrass-type approximation result holds between the finite memoryless polynomic functions and the memoryless continuous functions. In order to present the counterexample it is necessary to prove that certain causality properties of a multipower operator hold if and only if the symmetric multilinear generator of the multipower operator also has these properties in each linear argument. These results are important in their own right. Because this study is motivated by and supportive of [l], we shall adopt the notation, definitions, and conventions of this reference. In brief, H is a Hilbert space and W: H” -+ H is a multilinear operator if W[x, , x2 ,..., x,J is linear in each argument. W is symmetric if it is invariant under all possible permutations of arguments, for instance, W[xl , x2 ,... ] = W[x, , x1 ,... ] all x1 , x2 E H. The multilinear function W generates a multipower function, WI: H + H, by the formula IV(X) = W[x, x,..., x]. I f W generates @then there is a symmetric w determined by W which also generates l?’ (see [I, 2,4]). For this reason, we focus exclusively on multipower functions and their symmetric multilinear generators. A family {P : t E V} of orthoprojectors on H is said to be a resolution of the identity if Y is a linearly ordered set with maximal and minimal elements t, , t, , respectively, and if (1) P(H) 3_ P(H) whenever t > I, and (2) P(H) = (O},