Journal of Operator Theory

Volume 33, Issue 2, Spring 1995 pp. 279-297.

Denseness of the generalized eigenvectors of a discrete operator in a Banach space

Authors: Janet Burgoyne
Author institution: Department of Mathematics and Computer Science, South Dakota School of Mines and Technology, Rapid City, South Dakota 57701, U.S.A.

Summary: Let T be a closed, densely defined, linear operator in a separable, reflexive Banach space X, and assume that there exists $\xi \in \rho (t)$ such that $R_\xi (T)$ is a compact operator whose approximation numbers are p-summable, $0 < p < \infty$. The operator T is a special type of discrete operator, a so-called $C_p^{(a)}$-discrete operator. Let $\overline {{\rm{sp}}} (T)$ be the smallest closed subspace of X containing the subspace spanned by the generalized eigenvectors of T. Sufficient conditions are introduced which guarantee $\overline {{\rm{sp}}} (T) = X$. These conditions require that $\left\| {R_\lambda (T)} \right\|$ exhibit the decay rate ${\rm{O}}(\left| \lambda \right|^N )$ on certain rays in the complex plane. This work generalizes past Hilbert space theory developed by Dunford and Schwartz.

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