张量对称类上诱导线性算子的数值半径

Numerical Radius of Induced Operators on Symmetry Classes of Tensors

设V是一个n维线性空间,V_x^m(G)为V上的张量对称类.A为V的线性算子T的矩阵,K(A)为V_x^m(G)上的诱导线性算子K(T)的矩阵.本文从K(A)的数值半径Υ(K(A))和可分数值半径Υ_x(K(A))定义出发,研究了Υ(K(A))、Υ_x(K(A))与范数||A||_p(1≤p≤2)、广义矩阵函数d_x^G(A)的关系,得到了它们之间的两个不等式.

Let V be an n-dimensional linear space and Vχ^m（G） be a subspace of ×^mV, called the symmetry class of tensors over V associated with G and χ. Suppose A is a matrix of the linear operator T acting on V and K（A） is a matrix of the induced operator K（T） acting on Vχ^m（G）. Form definition of the numerical radius τ（K（A）） and decomposable numerical radius τχ（A）, two matrix inequalities involving the numerical radius τ（K（A））, the decomposable numerical radius τχ（A）, the norm ｜｜A｜｜2 and the generalized matrix function dχ^G（A） are obtained.