The joint essential numerical range of operators: convexity and related results
Chi-Kwong Li1, Yiu-Tung Poon2 Studia Math. 194 (2009), 91-104
doi:10.4064/sm194-1-6 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.
