On the normalized numerical range

Ilya M. Spitkovsky, Andrei Florian Stoica

Research output: Contribution to journalArticle

Abstract

The normalized numerical range of an operator A is defined as the set FN(A) of all the values 〈Ax, x〉/||Ax|| attained by unit vectors x ∉ ker A. We prove that FN(A) is simply connected, establish conditions for it to be star-shaped with the center at zero, to be open, closed, and to have empty interior. For some classes of operators (weighted shifts, isometries, essentially Hermitian) the complete description of FN(A) is obtained.

Original languageEnglish (US)
Article number11-15
Pages (from-to)219-240
Number of pages22
JournalOperators and Matrices
Volume11
Issue number1
DOIs
StatePublished - Mar 2017

Keywords

  • Essentially Hermitian operator
  • Normalized numerical range
  • Numerical range
  • Partial isometry
  • Weighted shift

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory

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