Numerical Radius of Induced Operators on Symmetry Classes of Tensors

Abstract:

      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.