The Spectral Scale and the k-Numerical Range
| dc.creator | Akemann, Charles A. | |
| dc.creator | Anderson, Joel | |
| dc.date | 2001-06-30 | |
| dc.date.accessioned | 2026-07-07T04:42:26Z | |
| dc.date.available | 2026-07-07T04:42:26Z | |
| dc.description | Suppose that c is a linear operator acting on an n-dimensional complex Hilbert Space H, and let tau denote the normalized trace on B(H). Set b_1 = (c+c*)/2 and b_2 = (c-c*)/2i, and write B for the the spectral scale of {b_1, b_2} with respect to tau. We show that B contains full information about (W_k)(c), the k-numerical range of c for each k =1,...,n. We then use our previous work on spectral scales to prove several new facts about (W_k)(c). For example, we show in Theorem 3.4 that the point lambda is a singular point on the boundary of (W_k)(c) if and only if lambda is an isolated extreme point of (W_k)(c). In this case lambda = (n/k)tau(cz), where z is a central projection in in the algebra generated by b_1, b_2 and the identity. We show in Theorem 3.5, that c is normal if and only if (W_k)(c) is a polygon for each k. Finally, it is shown in Theorem 5.4 that the boundary of (W_k)(c) is the finite union of line segments and curved real analytic arcs. | |
| dc.description | 34 pages | |
| dc.identifier | https://arxiv.org/abs/math/0107002 | |
| dc.identifier | http://arxiv.org/abs/math/0107002 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/61777 | |
| dc.subject | Rings and Algebras | |
| dc.subject | Operator Algebras | |
| dc.subject | 15A60, 47A12 | |
| dc.title | The Spectral Scale and the k-Numerical Range | |
| dc.type | text |