New pairs of matrices with convex generalized numerical ranges

Abstract

In this article, we are going to search for $n\times n$ matrices $A$ and $B$ such that their generalized numerical range $$W_A(B)=\{tr(A{U}^{\ast }BU):U^{\ast }U=U{U}^{\ast }=I\}$$ is convex. More specifically, we consider $A=\hat{A}\oplus I_k$ and $B=\hat{B}\oplus I_k$ where $\hat{A}$ and $\hat{B}$ are $2\times 2$. If $W_A(B)=W_{\hat{A}}(\hat{B})$ then it is a convex set.