Article ID Journal Published Year Pages File Type
4603369 Linear Algebra and its Applications 2008 15 Pages PDF
Abstract

Given positive integers n and p, and a complex finite dimensional vector space V, we let Sn,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 Sn,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∈Sn,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.

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory