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.
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Lyubich, Y., Markus, A. Connectivity of Level Sets of Quadratic Forms and Hausdorff-Toeplitz Type Theorems. Positivity 1, 239–254 (1997). https://doi.org/10.1023/A:1009760027676
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DOI: https://doi.org/10.1023/A:1009760027676