Invariant subspaces for certain linear transformations on a space of tensors

Abstract

Suppose each ofm, n and k is a positive integer. If p > 1, then S(m,p) denotes a certain inner product space whose elements are the real-valued symmetric p-linear functions on E_m . If A ϵ0 S(m,n) and B ϵ S(m,k), then the tensor and symmetric products are denoted by A ⨷ B and A · B, respectively. If k > n, then by AB is meant the member of S(m,k — n) such that (AB)(x₁,x₂,…,x_k-n) = < A,B(x₁,x₂,…,x_k-n) > for all x₁,x₂,…,x_k-n in E_m . If A ϵ S(m,n) and p ≥ n, then the linear operators M_p and G_p are defined as follows: if B ϵ S(m,p) then M_p(B) = A(A · B) and G_p(B) = A(A · B). These operators originally appeared in [1], and certain properties of these operators, especially minimum eigenvalue estimates for M_P , have been crucial to results in partial differential equations appearing in the recent papers of Neuberger (see [1], [2], and [3]). In this paper it is shown that under certain conditions on A there exists a resolution of the identity on S(m,p) each of whose members commutes with M_p and G_p . This establishes a collection of invariant subspaces for M_p and G_p . Also presented are estimates for the minimum eigenvalue of the restriction of M_p to the various invarient subspaces. In most cases these estimates improve on the single estimate used in [2] and [3].