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

Abstract

For a positive trace-class operator C and a bounded operator A, we provide an explicit description of the closure of the orbit-closed C-numerical range of A in terms of those operators submajorized by C and the essential numerical range of A. This generalizes and subsumes recent work of Chan, Li and Poon for the k-numerical range, as well as some of our own previous work on the orbit-closed C-numerical range.

Keywords:

• Numerical range
• C-numerical range
• convex
• trace-class
• essential numerical range
• majorization
• submajorization
• weak* convergence

AMS CLASSIFICATIONS:

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

Disclosure statement

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

Notes

1 This definition of the k-numerical range is the one given by Chan, Li and Poon. However, the reader should be aware that there is another definition, differing only by a scaling factor: $\left\{\frac{1}{k}\mathrm{T}\mathrm{r}\left(PA\right)|P\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}-k\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right\}.$The definition given in this footnote is the one originally described by Halmos in [13]. Both definitions appear throughout the literature, so one always has to be careful to see which definition the authors use.

2 Poon also made this connection in the case when C is finite rank [14].

3 Note that λ(C) is not generally uniquely determined since there may be unequal eigenvalues with the same modulus. However, if C is positive, then λ(C) is uniquely determined.

4 This is a general fact: if X is positive and Y is selfadjoint with Y ≥ −X, then Tr(Y₋) ≤ Tr(X). Indeed, if P is the range projection of Y₋, then we have Y₋ = P( − Y)P ≤ PXP, hence Tr(Y₋) ≤ Tr(PXP) ≤ Tr(X).

5 If either or both of these eigenvalues are zero, then X′ has rank 1 or 0.