On the normalized numerical range

Ilya 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 1 2017

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Numerical Range
Weighted Shift
Unit vector
Operator
Isometry
Star
Interior
Closed
Zero
Class

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory

Cite this

On the normalized numerical range. / Spitkovsky, Ilya; Stoica, Andrei Florian.

In: Operators and Matrices, Vol. 11, No. 1, 11-15, 01.03.2017, p. 219-240.

Research output: Contribution to journalArticle

Spitkovsky, Ilya ; Stoica, Andrei Florian. / On the normalized numerical range. In: Operators and Matrices. 2017 ; Vol. 11, No. 1. pp. 219-240.
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