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

Abstract

Given positive integers n and p, and a complex finite dimensional vector space V, we let S n , p ( V ) denote the set of all functions from V × V × ⋯ × V -( n + p copies) to C that are linear and symmetric in the first n positions, and conjugate linear symmetric in the last p positions. Letting κ = min { n , p } we introduce twisted inner products, [ · , · ] s , t , 1 ⩽ s , t ⩽ κ , on S n , p ( V ) , and prove the monotonicity condition [ F , F ] s , t ⩾ [ F , F ] u , v is satisfied when s ⩽ u ⩽ κ , t ⩽ v ⩽ κ , and F ∈ S n , p ( V ) . Using the monotonicity condition, and the Cauchy-Schwartz inequality, we obtain as corollaries many known inequalities involving norms of symmetric multilinear functions, which in turn imply known inequalities involving permanents of positive semidefinite Hermitian matrices. New tensor and permanental inequalities are also presented. Applications to partial differential equations are indicated.