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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 ${\mathbf A} = (A_1, \dots, A_m)\in {\mathcal B}({\mathcal H})^m$, the joint $k$-numerical range $W_k({\mathbf A})$ is the set of vector $(α_1, \dots, α_m) \in{\mathbb C}^m$ such that $α_i = \sum_{j = 1}^k \langle A_ix_j, x_j\rangle$ 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 $\dim {\mathcal H}$.