## 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