An extension of the Hausdorff-Toeplitz theorem on the numerical range

Abstract:

Let ${\mathcal {H}_n}$ be the set of all $n \times n$ hermitian matrices and ${\mathcal {U}_n}$ the set of all $n \times n$ unitary matrices. For any $c = ({c_1}, \ldots ,{c_n}) \in {{\mathbf {R}}^n}$ and ${A_1}$, ${A_2}$, ${A_3} \in {\mathcal {H}_n}$, let $W({A_1},{A_2},{A_3})$ denote the set \[ \{ ({\operatorname {tr}}[c]U{A_1}{U^*},{\operatorname {tr}}[c]U{A_2}{U^*},{\operatorname {tr}}[c]U{A_3}{U^*}):U \in {\mathcal {U}_n}\} ,\] where $[c]$ is the diagonal matrix with ${c_1}, \ldots ,{c_n}$ as diagonal entries. In this present note, the authors prove that if $n > 2$, then ${W_c}({A_1},{A_2},{A_3})$ is always convex. Equivalent statements of this result, in terms of definiteness and inclusion relations, are also given. These results extend the theorems of Hausdorff-Toeplitz, Finsler and Westwick on numerical ranges, respectively.