一阶偏导算子的正交数值域为实值的充要条件

Necessary and Sufficient Conditions for the Numerical Range of Derivation to be Real

设A_J∈L(V),i=1,…,m,A_1=A_1…A_m为A_1,…A_m的张量积,称D(A_1,…,A_m)=A_1I…I+IA_2I…I+…+I…IA_m为■A_i的一阶偏导算子,它的正交数值域为(D(A_1,…,A_m))={sum from i=1 to m(A_jv_j,v_j)|(v_i,v_j)=δ_(ij),i,j=1,…,m}(要求m=≤n=dimV)。本文给出了(D(A_1,…,A_m))=0,(D(A_1,…,A+m))R及D(A_1,…A_m)为厄米特算子的充要条件。

For A_i∈L(V), i=1,…, m, 1et A_i denote the tensor product of A_1, …, A_m, D(A_1, …, A_m)=A_1 I…I+IA_2I…I+…+I…IA_m the derivative operator of A_i of degree 1 is called. The orthognal onumerical range of D(A_1,…,A_m) is the set W (D(A_1, …, A_m)){sum from i=1 to m(A_iv_i, v_i)|(v_i, v_j)=δ_(ij), i, j=1, …m}(here m≤n=dimV is needed). The paper gives the necessary and sufficient conditions for D(A_1, …, A_m) to satisfy W(D(A_1, …, A_m)=0, W(D(A_1,…, A_m)) R and to be hermitian operator, respectively.