Numerical ranges of composition operators

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

Composition operators on the Hilbert Hardy space of the unit disk are considered. The shape of their numerical range is determined in the case when the symbol of the composition operator is a monomial or an inner function fixing 0. Several results on the numerical range of composition operators of arbitrary symbol are obtained. It is proved that 1 is an extreme boundary point if and only if 0 is a fixed point of the symbol. If 0 is not a fixed point of the symbol, 1 is shown to be interior to the numerical range. Some composition operators whose symbol fixes 0 and has infinity norm less than 1 have closed numerical ranges in the shape of a cone-like figure, i.e., a closed convex region with a corner at 1, 0 in its interior, and no other corners. Compact composition operators induced by a univalent symbol whose fixed point is not 0 have numerical ranges without corners, except possibly a corner at 0.

Original languageEnglish (US)
Pages (from-to)61-74
Number of pages14
JournalLinear Algebra and Its Applications
Volume331
Issue number1-3
DOIs
StatePublished - Jul 1 2001

Fingerprint

Numerical Range
Composition Operator
Mathematical operators
Chemical analysis
Fixed point
Interior
Hilbert spaces
Inner Functions
Closed
Monomial
Compact Operator
Cones
Hardy Space
Unit Disk
Hilbert
Figure
Extremes
Cone
Infinity
If and only if

Keywords

  • 47A12
  • 47B38
  • Composition operator
  • Hardy space
  • Numerical range

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

Cite this

Numerical ranges of composition operators. / Matache, Valentin.

In: Linear Algebra and Its Applications, Vol. 331, No. 1-3, 01.07.2001, p. 61-74.

Research output: Contribution to journalArticle

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