Abstract
The following question is considered: What is the smallest number γ(k) with the property that for every family X 1,⋯,X k of k selfadjoint and linearly independent operators on a real or complex Hilbert space H there exists a subspace H 0⊂H of dimension γ(k) such that the compressions of X 1,⋯,X k to H 0 are still linearly independent? Upper and lower bounds for γ(k) are established for any k, and the exact value is found for k=2,3. It is also shown that the set of all γ(k)-dimensional subspaces H 0 with the desired property is open and dense in the respective Grassmannian. The k=3 case is used to prove that the ratio numerical range W(A/B) of a pair of operators on a Hilbert space either has a non-empty interior, or lies in a line or a circle.
Original language | English (US) |
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Pages (from-to) | 3757-3766 |
Number of pages | 10 |
Journal | Linear Algebra and Its Applications |
Volume | 436 |
Issue number | 9 |
DOIs | |
State | Published - May 1 2012 |
Keywords
- Compressions
- Generalized numerical range
- Linear independence
- Selfadjoint operators
ASJC Scopus subject areas
- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics