# Connectivity of Level Sets of Quadratic Forms and Hausdorff-Toeplitz Type Theorems

Connectivity of Level Sets of Quadratic Forms and Hausdorff-Toeplitz Type Theorems 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> = 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Connectivity of Level Sets of Quadratic Forms and Hausdorff-Toeplitz Type Theorems

Positivity, Volume 1 (3) – Oct 14, 2004
16 pages

1

Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1009760027676
Publisher site
See Article on Publisher Site

### 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> = 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.

### Journal

PositivitySpringer Journals

Published: Oct 14, 2004

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