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 ${ \mathbf A} = (A_1, \dots, A_m)\in { \mathcal B}({ \mathcal H})^m$, the joint $k$-numerical range $W_k({ \mathbf A})$ is the set of vector $(\alpha_1, \dots, \alpha_m) \in{ \mathbb C}^m$ such that $\alpha_i = \sum_{ j = 1}^k \langle A_ix_j, x_j\rangle$ for an orthonormal set $\{ x_1, \ldots, x_k\}$ in ${ \mathcal H}$. Geometrical properties of $W_k({ \mathbf A})$ and their relations with the algebraic properties of $\{ A_1, \dots, A_m\}$ are investigated in this paper. For example, conditions for $W_k({ \mathbf A})$ to be convex are studied. Descriptions are given for the closure of $W_k({ \mathbf A})$ and the closure of ${ \rm conv}\, W_k({ \mathbf A})$ in terms of the joint essential numerical range of ${ \mathbf A}$ for infinite dimensional operators $A_1, \dots, A_m$. Characterizations are obtained for $W_k({ \mathbf A})$ or ${ \rm conv}\, W_k({ \mathbf A})$ 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 $W_k({ \mathbf A}) = W_k(D_1, \dots, D_m)$. Similar results are obtained for ${ \bf A}$ 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 $\dim { \mathcal H}$.
Categories: math.FA