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.
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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:
The definition given in this footnote is the one originally described by Halmos in [Citation13]. 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 [Citation14].
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.