Completely continuous and related multilinear operators

Completely continuous multilinear operators are defined and their properties investigated. This class of operators is shown to form a closed multi-ideal. Unlike the linear case, compact multilinear operators need not be completely continuous. The completely continuous maps are shown to be the closure of a subspace of the finite rank operators. Hilbert-Schmidt operators are also considered. An application to finding error bounds for solutions of multipower equations is presented.