Abstract
In a complex Hilbert spaceX for an arbitrary operator polynomialL(λ) (λ ∈ C) of degreem the following theorem is proved. If the equation (L(λ)x, x)=0 hasm distinct roots at every pointx ∈X, ‖x‖=1, then there existm pairwise disjoint connected sets in C such that each set contains a root at everyx. The minimal distance between the roots is separated from zero under the same assumption on the discriminant and the leading coefficient of that equation.
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Lyubich, Y. Separation of roots of matrix and operator polynomials. Integr equ oper theory 29, 52–62 (1997). https://doi.org/10.1007/BF01191479
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DOI: https://doi.org/10.1007/BF01191479