Numerical radius inequalities for tensor product of operators

Abstract: The two well-known numerical radius inequalities for the tensor product $A \otimes B$ acting on $\mathbb{H} \otimes \mathbb{K}$, where $A$ and $B$ are bounded linear operators defined on complex Hilbert spaces $\mathbb{H} $ and $ \mathbb{K},$ respectively are, $ \frac{1}{2} |A||B| \leq w(A \otimes B) \leq |A||B| $ and $w(A)w(B) \leq w(A \otimes B) \leq \min { w(A) |B|, w(B) |A| }. $ In this article we develop new lower and upper bounds for the numerical radius $w(A \otimes B)$ of the tensor product $A \otimes B $ and study the equality conditions for those bounds.