The joint essential numerical range of operators: convexity and related results

Chi-Kwong Li ¹   ; Yiu-Tung Poon ²

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.

Studia Mathematica, Tome 194 (2009) no. 1, pp. 91-104

Résumé

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.

Détail

DOI : 10.4064/sm194-1-6

Keywords: joint numerical range joint essential numerical range m tuple self adjoint operators dots acting infinite dimensional hilbert space shown always convex admits many equivalent formulations particular fixed dots obtained intersection sets form mathop nolimits dots dots where * has finite rank moreover closure mathop nolimits always star shaped elements star centers although mathop nolimits usually convex analog separation theorem obtained namely element notin mathop nolimits there linear functional sup mathop nolimits tilde where tilde obtained perturbing components finite rank self adjoint operator other results extending those single operator obtained