The Toeplitz-Hausdorff and Spectral Inclusion Theorems for Linear Relations

Silas Luliro Kito*, Gerald Wanjala

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


The Toeplitz-Hausdorff Theorem which was established in 1918 asserts that the numerical range of an operator is always convex. We prove this theorem for the numerical range of a linear relation relative to another linear relation. We also show that the closure of this numerical range contains the relative spectrum for these two linear relations. The classical results for a single linear relation can be deduced from the results obtained here.

Original languageEnglish
Pages (from-to)1637-1645
Number of pages9
JournalInternational Journal of Mathematics and Computer Science
Issue number4
Publication statusPublished - 2021
Externally publishedYes


  • Convexity
  • Linear relation
  • Numerical range
  • Spectral inclusion

ASJC Scopus subject areas

  • Computer Science (miscellaneous)
  • Algebra and Number Theory
  • Statistics and Probability
  • Numerical Analysis
  • Modelling and Simulation
  • Discrete Mathematics and Combinatorics
  • Computational Mathematics
  • Applied Mathematics


Dive into the research topics of 'The Toeplitz-Hausdorff and Spectral Inclusion Theorems for Linear Relations'. Together they form a unique fingerprint.

Cite this