Numerical range of tensor product of operators in semi-Hilbert spaces

Abstract

Let $A$ and $B$ be two positive bounded linear operators acting on the complex Hilbert spaces $H$ and $K$, respectively. In this paper, we study the $(A\otimes B)$-numerical range $W_{A\otimes B}(T\otimes S)$ of the tensor product $T\otimes S$ for two bounded linear operators $T$ and $S$ on $H$ and $K$, respectively. In the context of this work, we demonstrate that if either $T$ is $A$-hyponormal or $S$ is $B$-hyponormal, then $\overline{W_{A\otimes B}(T\otimes S)}=\mathrm{co}\left(\overline{W_A\left(T\right)}\cdot \overline{W_B\left(S\right)}\right),$where $W_A\left(T\right)$ and $W_B\left(S\right)$ denote the $A$-numerical range of $T$ and the $B$-numerical range of $S$, respectively. Here, $\mathrm{co}\left(\cdot \right)$ and the over-line denote the convex hull and the closure, respectively. Moreover, we provide some $(A\otimes B)$-numerical radius inequalities.