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 | |
| 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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