Compressions of linearly independent selfadjoint operators

Leiba Rodman, Ilya M. Spitkovsky

Research output: Contribution to journalArticlepeer-review

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 languageEnglish (US)
Pages (from-to)3757-3766
Number of pages10
JournalLinear Algebra and Its Applications
Volume436
Issue number9
DOIs
StatePublished - 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

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