Connectivity of level sets of quadratic forms and Hausdorff-Toeplitz type theorems

Yuri Lyubich, Alexander Markus

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

Main theorem: for an arbitrary linear operator A : X → X in a complex pre-Hilbert space X, dim X ≥ 3, all level sets {x ∈ X : (Ax, x) = λ, ∥x∥ = 1} are connected. This fails if dim X = 2 and λ ∈ int W (A) where W (A) is the numerical range. The main theorem implies the known result on convexity of generalized numerical range of three Hermitian operators.

Original languageEnglish
Pages (from-to)239-254
Number of pages16
JournalPositivity
Volume1
Issue number3
DOIs
StatePublished - 1 Jan 1997

Keywords

  • Connectivity
  • Convexity
  • Generalized numerical range
  • Hausdorff-Toeplitz theorem
  • Quadratic forms

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Analysis
  • Mathematics (all)

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