The joint $k$-numerical range of operators

Let $\mathcal{B}(\mathcal{H})$ be the algebra of all bounded linear operators on the Hilbert space $\mathcal{H}$. For a positive integer $k$ less than the dimension of $\mathcal{H}$ and ${\mathbf A} = (A_1, \dots, A_m)\in {\mathcal B}({\mathcal H})^m$,thejoint$k$-numericalrange$W_k({\mathbf A})$isthesetofvector$(\alpha_1, \dots, \alpha_m) \in{\mathbb C}^m$suchthat$\alpha_i = \sum_{j = 1}^k \langle A_ix_j, x_j\rangle$foranorthonormalset${x_1, \ldots, x_k}$in${\mathcal H}$\mathrm{.}Geometricalpropertiesof$W_k({\mathbf A})$andtheirrelationswiththealgebraicpropertiesof${A_1, \dots, A_m}$areinvestigatedinthispaper\mathrm{.}Forexample,conditionsfor$W_k({\mathbf A})$ to be convex are studied. Descriptions are given for the closure of $W_k(\mathbf{A})$ and the closure of ${\rm conv}, W_k({\mathbf A})$intermsofthejointessentialnumericalrangeof${\mathbf A}$ for infinite dimensional operators $A_1,\dots ,A_m$. Characterizations are obtained for $W_k(\mathbf{A})$ or $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}{W}_k(\mathbf{A})$ to be closed. It is shown that $W_k(\mathbf{A})$ is a polyhedral set if and only if $A_1,\dots ,A_k$ have a common reducing subspace $\mathbf{V}$ of finite dimension such that the compression of $A_1,\dots ,A_m$ on the subspace $\mathbf{V}$ are diagonal operators $D_1,\dots ,D_m$ and $W_k({\mathbf A}) = W_k(D_1, \dots, D_m)$\mathrm{.}Similarresultsareobtainedfor${\bf A}$ such that the closure of $W_k(\mathbf{A})$ is polyhedral. Classifications are given for operators satisfying (1) $\{A_1,\dots ,A_m\}$ is a commuting family of normal operators, or (2) $W_k(A_1,\dots ,A_m)$ is polyhedral for every positive integer $k$ less than $\dim \mathcal{H}$.