On the normalized numerical range

Ilya M. Spitkovsky, Andrei Florian Stoica

Research output: Contribution to journalArticlepeer-review


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
Issue number1
StatePublished - Mar 2017


  • 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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