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

1. Department of Mathematics
   The College of William and Mary
   Williamsburg, VA 23185, U.S.A.
   
2. Department of Mathematics
   Iowa State University
   Ames, IA 50011, U.S.A.