Denseness of the generalized eigenvectors of an H-S discrete operator
Let T be a closed densely defined linear operator in a Hilbert space H. Assume that there exists a point \(\xi\in \rho (T)\) such that the resolvent \(R_{\xi}(T)\) is a Hilbert-Schmidt oprator. Then, T is called an H-S discrete operator. It is a result of Dunford and Schwartz that if \(R_{\lambda}(T)\) satisfies \(\| R_{\lambda}(T)\| =O(| \lambda |^{-N})\) as \(\lambda\) \(\to \infty\) along certain rays \(\gamma_ 1,...,\gamma_ 5\), then H is the closure of the subspace of H spanned by the generalized eigenvectors of T. In particular, this result applies if T is the differential operator in \(L^ 2[0,1]\) determined by \(\tau =-[d/dt]^ 2\) and regular boundary conditions. In the present paper the authors generalize the Dunford and Schwartz result so that irregular boundary conditions are covered. Their main theorem is the following: Let T be an H-S discrete operator. Let \(\sigma (T)=\{\lambda_ i\}^{\infty}_{i=1}\). Assume that there exists a set of five rays such that (i) the angles between adjacent rays are less than \(\pi\) /2; (ii) for \(| \lambda |\) sufficiently large all the points on the five rays belong to \(\rho\) (T), and \(\| R_{\lambda}(T)\|\) is bounded for these \(\lambda\) ; (iii) on at least one of the rays \(\| R_{\lambda}(T)\| \to 0\) as \(\lambda\) \(\to \infty.\) Then \(\overline{S_{\infty}}=\overline{S^*_{\infty}}=H\) and \(M_{\infty}=M^*_{\infty}=\{0\}.\) Here \(S_{\infty}\) and \(M_{\infty}\) are the subspaces of H consisting of all \(x\in H\) such that \(x=\sum^{\infty}_{i=1}P_ ix\) and such that \(P_ ix=0\) for \(i=1,2,...\), respectively. \(P_ i\) is the projection of H onto the generalized eigenspace \({\mathcal N}([\lambda_ iI- T]^{m_ i})\) along \({\mathcal R}([\lambda_ iI-T]^{m_ i}).\) \(m_ i\) is the ascent of \(\lambda_ iI-T\). \(S^*_{\infty}\) and \(M^*_{\infty}\) are defined similarly using \(P^*_ i\).
closed densely defined linear operator in a Hilbert space
scientific article; zbMATH DE number 4113336