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Abstract

Let $W({\bf A})$ and $W_{\rm e}({\bf A})$ be the joint numerical range and the joint essential numerical range of an $m$-tuple of self-adjoint operators ${\bf A} = (A_1, \dots, A_m)$ acting on an infinite-dimensional Hilbert space. It is shown that $W_{\rm e}({\bf A})$ is always convex and admits many equivalent formulations. In particular, for any fixed $i \in \{1, \dots, m\}$, $W_{\rm e}({\bf A})$ can be obtained as the intersection of all sets of the form $$\mathop{\bf cl}\nolimits(W(A_1, \dots, A_{i+1}, A_i+F, A_{i+1}, \dots, A_m)),$$ where $F = F^*$ has finite rank. Moreover, the closure $\mathop{\bf cl}\nolimits(W({\bf A}))$ of $W({\bf A})$ is always star-shaped with the elements in $W_{\rm e}({\bf A})$ as star centers. Although $\mathop{\bf cl}\nolimits(W({\bf A}))$ is usually not convex, an analog of the separation theorem is obtained, namely, for any element ${\bf d} \notin \mathop{\bf cl}\nolimits(W({\bf A}))$, there is a linear functional $f$ such that $f({\bf d}) > \sup\{ f({\bf a}): {\bf a}\in \mathop{\bf cl}\nolimits (W( \tilde {\bf A}) )\},$ where $\tilde {\bf A}$ is obtained from ${\bf A}$ by perturbing one of the components $A_i$ by a finite rank self-adjoint operator. Other results on $W({\bf A})$ and $W_{\rm e}({\bf A})$ extending those on a single operator are obtained.