Twisted inner products and contraction inequalities on spaces of contravariant and covariant tensors (Q947611)

The objective of this paper is to define a new kind of inner product on the space of tensors and discuss some of its properties. Given positive integers \(n\) and \(p\), and a complex finite dimensional vector space \(V\), let \(\mathbf{S}_{n,p}(V)\) denote the set of all functions from \(V \times V \times \cdots \times V\) - (\(n + p\) copies) to \(\mathbb{C}\) that are linear and symmetric in the first \(n\) positions, and conjugate linear symmetric in the last \(p\) positions. Let \(\kappa = \min\{n,p\}\). For each integer pair \((s,t)\) such that \(0\leq s,t \leq \kappa\), the author defined a sesquilinear form \([\cdot,\cdot]_{s,t}\) on \(\mathbf{S}_{n,p}(V)\) and proved that each of the \([\cdot,\cdot]_{s,t}\) is an inner product on \(\mathbf{S}_{n,p}(V)\). It is called twisted inner product. It is proved that the monotonicity condition \([F,F]_{s,t} \geq [F,F]_{u,v}\) is satisfied then \(s\leq u\leq \kappa\) and \(t\leq v\leq \kappa\), and \(F\in \mathbf{S}_{n,p}(V)\). Using this monotonicity condition and the Cauchy-Schwartz inequality, many inequalities involving norms of symmetric multilinear functions are proved, and then applied to interesting cases.