Multilinear operator ideals associated to Saphar type \((n+1)\)-tensor norms
Let \(1 \leq p \leq \infty\) and let \(p'\) be the conjugate exponent such that \(1/p + 1/p' = 1\). The definition of the classical Saphar \(p\)-tensor norm involves \(p\) weakly and \(p'\) strongly summable sequences. Let \(n \geq 1\) and \(1 \leq k \leq n+1\). The purpose of this paper is to study an \((n+1)\)-fold tensor norm \(g_{r,s}\) naturally extending the well-known tensor norm \(g_p\) of Saphar. This new tensor norm on the tensor product \(\bigotimes_{j=1}^{n+1} E_j\) of \(n+1\) normed spaces \(E_j\) depends on two vector parameters \(r=(r_1,r_2,\dots,r_k)\) and \(s=(r_{k+1},r_{k+1},\dots,r_{n+1})\). The corresponding sequences of conjugate exponents are \(r' = (r'_1,r'_2,\dots,r'_k)\) and \(s' = (r'_{k+1},r'_{k+2},\dots,r'_{n+1})\). These sequences are required to satisfy \[ 1 = \frac{1}{r_1} + \frac{1}{r_1} + \cdots + \frac{1}{r_k} + \frac{1}{r'_{k+1}} + \frac{1}{r'_{k+2}} + \cdots + \frac{1}{r'_{n+1}}. \] As in the classical case, the definition of the norm involves \(r_j\) strongly and \(r'_j\) weakly summable sequences. The dual of the tensor product with this norm is identified with certain multilinear maps called \(t\)-absolutely summing maps. The elements in the completion of the \(g_{r,s}\) tensor product lead to multilinear \((r,s)\)-nuclear operators from \(\prod_{j=1}^n E_j\) into \(E_{n+1}\). The multilinear operator ideal associated with the \(g_{r,s}\)-tensor norm, the \((r,s)\)-integral operators, is also characterized.
\(g_{r,s}\)-nuclear and \(g_{r,s}\)-integral \(n\)-linear operators
\(t\)-absolutely summing \(n\)-linear operators
scientific article; zbMATH DE number 5156945