Connectivity of level sets of quadratic forms and Hausdorff-Toeplitz type theorems

Yuri Lyubich; Alexander Markus

doi:10.1023/A:1009760027676

Connectivity of level sets of quadratic forms and Hausdorff-Toeplitz type theorems

Yuri Lyubich, Alexander Markus

Department of Mathematics

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

6 Scopus citations

Abstract

Main theorem: for an arbitrary linear operator A : X → X in a complex pre-Hilbert space X, dim X ≥ 3, all level sets {x ∈ X : (Ax, x) = λ, ∥x∥ = 1} are connected. This fails if dim X = 2 and λ ∈ int W (A) where W (A) is the numerical range. The main theorem implies the known result on convexity of generalized numerical range of three Hermitian operators.

Original language	English
Pages (from-to)	239-254
Number of pages	16
Journal	Positivity
Volume	1
Issue number	3
DOIs	https://doi.org/10.1023/A:1009760027676
State	Published - 1 Jan 1997

Keywords

Connectivity
Convexity
Generalized numerical range
Hausdorff-Toeplitz theorem
Quadratic forms

ASJC Scopus subject areas

Theoretical Computer Science
Analysis
General Mathematics

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abstract = "Main theorem: for an arbitrary linear operator A : X → X in a complex pre-Hilbert space X, dim X ≥ 3, all level sets {x ∈ X : (Ax, x) = λ, ∥x∥ = 1} are connected. This fails if dim X = 2 and λ ∈ int W (A) where W (A) is the numerical range. The main theorem implies the known result on convexity of generalized numerical range of three Hermitian operators.",

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N2 - Main theorem: for an arbitrary linear operator A : X → X in a complex pre-Hilbert space X, dim X ≥ 3, all level sets {x ∈ X : (Ax, x) = λ, ∥x∥ = 1} are connected. This fails if dim X = 2 and λ ∈ int W (A) where W (A) is the numerical range. The main theorem implies the known result on convexity of generalized numerical range of three Hermitian operators.

AB - Main theorem: for an arbitrary linear operator A : X → X in a complex pre-Hilbert space X, dim X ≥ 3, all level sets {x ∈ X : (Ax, x) = λ, ∥x∥ = 1} are connected. This fails if dim X = 2 and λ ∈ int W (A) where W (A) is the numerical range. The main theorem implies the known result on convexity of generalized numerical range of three Hermitian operators.

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KW - Convexity

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Connectivity of level sets of quadratic forms and Hausdorff-Toeplitz type theorems

Abstract

Keywords

ASJC Scopus subject areas

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