The basis property of the root vectors of weakly perturbed operators (Q1086805)

Sufficient conditions are presented for the eigenvectors and dual eigenvectors of a weakly perturbed selfadjoint operator with discrete spectrum to form a Riesz basis. Let L be a positive selfadjoint operator on a Hilbert space with discrete spectrum and let T be a closed operator such that \(D_ L\subseteq D_ T\), \(\| Tx\| \leq C\| L^{\beta}x\|\), \(x\in D_ L\), \(\limsup_{t\to \infty}N(t,L)r^{- \alpha}<\infty\) for some \(\alpha,\beta,0\leq \beta <1\), \(0<\alpha \leq 1- \beta\) where N(t,L) is the number of eigenvalues of L less than t. The paper deals with the perturbed operator \(A=L+T\). The eigenvalues of A are subjected to the condition that numbers d, p exist such that for any real number \(\lambda >0\) the number of eigenvalues \(\lambda_ n\) with real part \(\alpha_ n=Re \lambda_ n\) within d of \(\lambda\) is less than p, i.e. \(\sum_{| \alpha_ n-\lambda | <d}1<p\). Assuming this condition to be satisfied it is proved that the biorthgonal system of eigenvectors \(\{e_ n\}\), \(\{g_ n\}\) of A and \(A^*\) form a Riesz basis if and only if \(\sup \| e_ n\| \leq M_ 1<\infty\), \(\sup \| g_ n\| \leq M_ 2<\infty\).