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1. CMB 1999 (vol 42 pp. 104)

Nikolskaia, Ludmila
Instabilité de vecteurs propres d'opérateurs linéaires
We consider some geometric properties of eigenvectors of linear operators on infinite dimensional Hilbert space. It is proved that the property of a family of vectors $(x_n)$ to be eigenvectors $Tx_n= \lambda_n x_n$ ($\lambda_n \noteq \lambda_k$ for $n\noteq k$) of a bounded operator $T$ (admissibility property) is very instable with respect to additive and linear perturbations. For instance, (1)~for the sequence $(x_n+\epsilon_n v_n)_{n\geq k(\epsilon)}$ to be admissible for every admissible $(x_n)$ and for a suitable choice of small numbers $\epsilon_n\noteq 0$ it is necessary and sufficient that the perturbation sequence be eventually scalar: there exist $\gamma_n\in \C$ such that $v_n= \gamma_n v_{k}$ for $n\geq k$ (Theorem~2); (2)~for a bounded operator $A$ to transform admissible families $(x_n)$ into admissible families $(Ax_n)$ it is necessary and sufficient that $A$ be left invertible (Theorem~4).

Keywords:eigenvectors, minimal families, reproducing kernels
Categories:47A10, 46B15

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