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

Closedness of the orbit-closed $C$-numerical range and submajorization

“…Since K is separable, this topology is metrizable on trace-norm bounded sets (by Banach-Alaoglu) and weak * convergence This concludes the necessary notation. We now review some background material which will be necessary, most of which comes from [7]. Our main result from [7] is:…”

“…We now review some background material which will be necessary, most of which comes from [7]. Our main result from [7] is:…”

“…• 1 . We have neglected mentioning this C-numerical range of Dirr and von Ende primarily because, for reasons discussed in [7], we feel that W O(C) (A) is actually the more natural extension to C infinite rank of the (previously existing, even as early as 1975 in [8]) definition for C finite rank. The next example reinforces this sentiment by establishing that the second equality in Theorem 3.4 does not hold if one replaces W O(C) (A) with W C (A) everywhere, despite the fact that (by [7,Theorem 3.1])…”

“…In our recent paper [7], we introduced the orbit-closed C-numerical range for C ∈ L 1 , the trace class, which is defined as follows. Let U (C) denote the unitary orbit of C under the natural conjugation action of the unitary group.…”

“…Our main theorem (Theorem 3.4) generalizes (1.3) and provides an independent proof of this fact. Moreover, as we showed in [7] that W O(C) (A) is intimately connected with majorization 2 (denoted ≺, see Definition 2.1) when C is selfadjoint, so also we connect the closure W O(C) (A) to submajorization (denoted Î, see Definition 2.1). In particular, we prove in Theorem 3.4, for C ∈ L + 1 and A ∈ B(H),…”