Invariant subspaces for certain linear transformations on a space of tensors
Titel:
Invariant subspaces for certain linear transformations on a space of tensors
Auteur:
Pate, Thomas H.
Verschenen in:
Linear & multilinear algebra
Paginering:
Jaargang 5 (1977) nr. 1 pagina's 11-18
Jaar:
1977
Inhoud:
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 Em. If A ε0 S(m,n) and B ε S(m,k), then the tensor and symmetric products are denoted by A O× 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,…,xk-n) = < A,B(x1,x2,…,xk-n) > for all x1,x2,…,xk-n in Em. If A ε S(m,n) and p ≥ n, then the linear operators Mp and Gp are defined as follows: if B ε S(m,p) then Mp(B) = A(A · B) and Gp(B) = A(A · B). These operators originally appeared in [1], and certain properties of these operators, especially minimum eigenvalue estimates for MP, 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 Mp and Gp. This establishes a collection of invariant subspaces for Mp and Gp. Also presented are estimates for the minimum eigenvalue of the restriction of Mp to the various invarient subspaces. In most cases these estimates improve on the single estimate used in [2] and [3].
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