The Toeplitz-Hausdorff and Spectral Inclusion Theorems for Linear Relations

Silas Luliro Kito; Gerald Wanjala

The Toeplitz-Hausdorff and Spectral Inclusion Theorems for Linear Relations

Silas Luliro Kito^*, Gerald Wanjala

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

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 language	English
Pages (from-to)	1637-1645
Number of pages	9
Journal	International Journal of Mathematics and Computer Science
Volume	16
Issue number	4
Publication status	Published - 2021
Externally published	Yes

Keywords

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

Cite this

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title = "The Toeplitz-Hausdorff and Spectral Inclusion Theorems for Linear Relations",

abstract = "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.",

keywords = "Convexity, Linear relation, Numerical range, Spectral inclusion",

author = "Kito, {Silas Luliro} and Gerald Wanjala",

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PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Convexity

KW - Linear relation

KW - Numerical range

KW - Spectral inclusion

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ER -

The Toeplitz-Hausdorff and Spectral Inclusion Theorems for Linear Relations

Abstract

Keywords

ASJC Scopus subject areas

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