Geodesic
Instabilité de vecteurs propres d’opérateurs linéaires
Canadian mathematical bulletin, Tome 42 (1999) no. 1, pp. 104-117

Voir la notice de l'article provenant de la source Cambridge University Press

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 $({{x}_{\text{n}}})$ to be eigenvectors $T{{x}_{n}}={{\lambda }_{n}}{{x}_{n}}({{\lambda }_{n}}\ne {{\lambda }_{k}}\text{for}\,n\ne k)$ of a bounded operator $T$ (admissibility property) is very instable with respect to additive and linear perturbations. For instance, (1) for the sequence ${{\left( {{x}_{n}}+{{\epsilon }_{n}}{{v}_{n}} \right)}_{n\ge k(\epsilon )}}$ to be admissible for every admissible $({{x}_{\text{n}}})$ and for a suitable choice of small numbers ${{\epsilon }_{n}}\,\ne \,0$ it is necessary and sufficient that the perturbation sequence be eventually scalar: there exist ${{\text{ }\!\!\gamma\!\!\text{ }}_{n}}\,\in \,C$ such that ${{v}_{n}}\,=\,{{\gamma }_{n}}{{v}_{k}}\,\text{for}\,n\,\ge \,\text{k}$ (Theorem 2); (2) for a bounded operator $A$ to transform admissible families $({{x}_{\text{n}}})$ into admissible families $(A{{x}_{n}})$ it is necessary and sufficient that $A$ be left invertible (Theorem 4).
DOI : 10.4153/CMB-1999-012-0
Mots-clés : 47A10, 46B15, eigenvectors, minimal families, reproducing kernels 
Nikolskaia, Ludmila. Instabilité de vecteurs propres d’opérateurs linéaires. Canadian mathematical bulletin, Tome 42 (1999) no. 1, pp. 104-117. doi: 10.4153/CMB-1999-012-0
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