Convexity of the orbit-closed C-numerical range and majorization

Abstract

We introduce and investigate the orbit-closed C-numerical range, a natural modification of the C-numerical range of an operator introduced for C trace-class by Dirr and vom Ende. Our orbit-closed C-numerical range is a conservative modification of theirs because these two sets have the same closure and even coincide when C is finite rank. Since Dirr and vom Ende's results concerning the C-numerical range depend only on its closure, our orbit-closed C-numerical range inherits these properties, but we also establish more. For C self-adjoint, Dirr and vom Ende were only able to prove that the closure of their C-numerical range is convex and asked whether it is convex without taking the closure. We establish the convexity of the orbit-closed C-numerical range for self-adjoint C without taking the closure by providing a characterization in terms of majorization, unlocking the door to a plethora of results which generalize properties of the C-numerical range known in finite dimensions or when C has finite rank. Under rather special hypotheses on the operators, we also show the C-numerical range is convex, thereby providing a partial answer to the question posed by Dirr and vom Ende.

Keywords:

• Numerical range
• C-numerical range
• majorization
• convex
• trace-class
• Toeplitz–Hausdorff theorem
• unitary orbit
• Hausdorff distance
• essential spectrum

AMS Classifications:

• Primary 47A12
• 47B15
• Secondary 52A10
• 52A40
• 26D15

Acknowledgments

The authors thank Gary Weiss for providing helpful suggestions in the draft of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Or only slightly more generally, C normal with collinear eigenvalues. In this paper, we have many results for self-adjoint C, but they generally have trivial unmentioned corollaries for C normal with collinear eigenvalues by means of Proposition 3.2(iii). We neglect these slightly more general statements in favor of the self-adjoint ones solely for clarity and simplicity of exposition.

2 Although Poon's paper addresses the finite-dimensional situation, the finite rank case easily reduces to the finite-dimensional case, as Westwick himself employed.

3 In [18], this is actually stated in terms of the so-called partial isometry orbit, $\mathcal{V}\left(C\right)$, but [19, Proposition 2.1.12] guarantees that ${\overline{\mathcal{U}\left(C\right)}}^{\parallel \mathbf{\cdot }\parallel }=\mathcal{V}\left(C\right)$ for $C\in {\mathcal{K}}^{+}$.

4 In the case of II₁ factors, the analogous notion is the spectral scale.

5 Recall that a supporting line L for a convex set C in the plane is a line such that $L\cap \overline{C}\ne set$ and C is entirely contained within one of the closed half-planes determined by L. Notice that this latter condition ensures $L\cap \overline{C}\subseteq \mathrm{\partial }C$.

6 This result is actually much stronger than we need because it provides tight bounds on the required size of the norm $\parallel B_{\theta }\parallel$. For our purposes, the result we need could be obtained by straightforward, albeit somewhat tedious, arguments using the continuous functional calculus.

7 If $\dim \ker C=\dim \ker A$, then $X:={\mathbf{0}}_{{\ker }_A}\oplus \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(s\right(C\left)\right)\in \mathcal{U}\left(C\right)$, and $A={\mathbf{0}}_{\ker A}\oplus \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(s\right(A\left)\right)$ relative to the proper basis so $\mathrm{T}\mathrm{r}\left(XA\right)=sup{W}_{\mathcal{O}\left(C\right)}\left(A\right)=sup{W}_C\left(A\right)$ by Proposition 5.1.

8 This notation is common in the later literature, but Marcus actually used the notation P_C(A) to refer to the convex hull of the C-spectrum, and he called this the C-eigenpolygon.

9 In [18], this is stated in terms of the so-called partial isometry orbit, $\mathcal{V}\left(C\right)$, but [19, Proposition 2.1.12] guarantees that ${\overline{\mathcal{U}\left(C\right)}}^{\parallel \mathbf{\cdot }\parallel }=\mathcal{V}\left(C\right)$ for $C\in {\mathcal{K}}^{+}$.

10 In the case when $\ker C$ is nontrivial but finite dimensional, the first author has conjectured a characterization of $E\left(\mathcal{U}\right(C\left)\right)$ and has established that this conjectured set is convex. See [40, Conjecture 3.6, Lemma 4.2] for details.