Extensions of the Hilbert-multi-norm in Hilbert C*-modules

( ) Dales and Polyakov introduced a multi-norm (parallel to.parallel to n((2,2)) : n is an element of N) based on a Banach space X and showed that it is equal with the Hilbert-multi-norm (parallel to.parallel to n(H) : n is an element of N) based on an infinite-dimensional Hilbert space H . We enrich the theory and present three extensions of the Hilbert-multi-norm for a Hilbert C*-module H . We denote these multi-norms by (parallel to.parallel to n(H) : n is an element of N), ((parallel to.parallel to n(*) : n is an element of N), and (parallel to.parallel to(P(u))(n) : n is an element of N). We show that?x?P(A) n >= ?x?Xn <= ?x?& lowast;nfor each x is an element of X n.In the casewhenXis a Hilbert K (H)-module, for each x is an element of Xn, we observe that ?middot?Pn(A) = ?middot?Xn. Furthermore, if His separable and X is infinite-dimensional, we prove that ?x?X n = ?x?& lowast;n. Among other things, we show that small and orthogonal decompositions with respect to these multi-norms are equivalent. Several examples are given to support the new findings.