Let $T$ be a selfadjoint bounded operator acting in a real Hilbert space $H$, and denote by $S$ the unit sphere of $H$. Assume that $l_0$ is an isolated eigenvalue of $T$ of odd multiplicity greater than $1$. Given an arbitrary operator $B cl H o H$ of class $Csp{1}$, we prove that for any $e e 0$ sufficiently small there exists $x_e in S$ and $l_e$ near $l_0$, such that $Tx_e + e B(x_e) = l_e x_e$. This result was conjectured, but not proved, in a previous article by the authors. We provide an example showing that the assumption that the multiplicity of $l_0$ is odd cannot be removed.
Topological Persistence of the Normalized Eigenvectors of a Perturbed Selfadjoint Operator / R. Chiappinelli; M. Furi; M.P. Pera. - In: APPLIED MATHEMATICS LETTERS. - ISSN 0893-9659. - STAMPA. - 2010:(2010), pp. 193-197. [10.1016/j.aml.2009.09.011]
Topological Persistence of the Normalized Eigenvectors of a Perturbed Selfadjoint Operator
FURI, MASSIMO;PERA, MARIA PATRIZIA
2010
Abstract
Let $T$ be a selfadjoint bounded operator acting in a real Hilbert space $H$, and denote by $S$ the unit sphere of $H$. Assume that $l_0$ is an isolated eigenvalue of $T$ of odd multiplicity greater than $1$. Given an arbitrary operator $B cl H o H$ of class $Csp{1}$, we prove that for any $e e 0$ sufficiently small there exists $x_e in S$ and $l_e$ near $l_0$, such that $Tx_e + e B(x_e) = l_e x_e$. This result was conjectured, but not proved, in a previous article by the authors. We provide an example showing that the assumption that the multiplicity of $l_0$ is odd cannot be removed.File | Dimensione | Formato | |
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