The perturbation theory for linear operators of discrete type (Q1075573)

A perturbation result for unbounded operators of discrete type [\textit{N. Dunford} and \textit{J. Schwartz}, Linear Operators, part III (1971; Zbl 0243.47001): Spectral Operators, XIX, 2, Theorem 7] is derived using the notion of unconditional basis instead of assuming weak completeness of the underlying Banach space. The following definition is the starting point for the generalization of a theorem of J. Schwartz und H. P. Kramer: A linear operator T in a Banach space B is called of discrete type, if \(\rho\) (T)\(\neq \emptyset\) and there exist an unconditional basis \(\{x_ n\}\) of B, a sequence of complex numbers \(\{\lambda_ n\}\) and a positive integer N, such that \(| \lambda_ n| \to \infty\) \(\lambda_ n\neq \lambda_ m\), n,m\(\in {\mathbb{N}}\), \(m>N\), \(n\neq m\), \(Tx_ n=\lambda_ nx_ n\), \(n>N\), \(T[x_ 1,...,x_ N]\subset [x_ 1,...,x_ N]\) and \(\sigma (T|_{[x_ 1,...,x_ N]})=\{\lambda_ 1,...,\lambda_ N\}\).