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

中文关键词:  张量对称类  诱导线性算子  数值半径.

英文关键词:symmetry class of tensors  induced operator  numerical radius.

中文摘要:

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

英文摘要:

      Let $V$ be an $n$-dimensional linear space and $V_\chi^m(G)$ be a subspace of $\otimes^m V$, called the symmetry class of tensors over $V$ associated with $G$ and $\chi$. 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 $V_\chi^m(G)$. Form definition of the numerical radius $r(K(A))$ and decomposable numerical radius $r_\chi(A)$, two matrix inequalities involving the numerical radius $r(K(A))$, the decomposable numerical radius $r_\chi(A)$, the norm $\|A\|_2$ and the generalized matrix function $d_\chi^G(A)$ are obtained.