Proceedings of the American Mathematical Society, 1991
In this paper, upper bounds for the difference between the eigenvalues and the eigenvectors of a ... more In this paper, upper bounds for the difference between the eigenvalues and the eigenvectors of a closed linear operator D D and those of D + F D + F , where F F is a bounded linear operator, are given in terms of the norm of F F . These results are applied to approximate the eigenvalues and the eigenvectors of a diagonally infinite matrix by those of its corresponding diagonal matrix.
Recently A. Gutek, D. Hart, J. Jamison and M. Rajagopalan have obtained many significiant results... more Recently A. Gutek, D. Hart, J. Jamison and M. Rajagopalan have obtained many significiant results concerning shift operators on Banach spaces. Using a result of Holsztynski they classify isometric shift operators on C(X) for any compact Hausdorff space X into two (not necessarily disjoint) classes. If there exists an isometric shift operator T: C(X) → C(X) of type II, they show that X is necessarily separable. In case T is of type I, they exhibit a paticular infinite countable set of isolated points in X. Under the additional assumption that the linear functional Γ carrying f ∊ C(X) to Tf(p) ∊ is identically zero, they show that D is dense in X. They raise the question whether D will still be dense in X even when Γ ≠0. In this paper we give a negative answer to this question. In fact, given any integer l ≥ 1, we construct an example of an isometric shift operator T: C(X) —> C(X) of type I with X \ having exactly / elements, where is the closure of D in X.
Proceedings of the American Mathematical Society, 1991
In this paper, upper bounds for the difference between the eigenvalues and the eigenvectors of a ... more In this paper, upper bounds for the difference between the eigenvalues and the eigenvectors of a closed linear operator D D and those of D + F D + F , where F F is a bounded linear operator, are given in terms of the norm of F F . These results are applied to approximate the eigenvalues and the eigenvectors of a diagonally infinite matrix by those of its corresponding diagonal matrix.
Recently A. Gutek, D. Hart, J. Jamison and M. Rajagopalan have obtained many significiant results... more Recently A. Gutek, D. Hart, J. Jamison and M. Rajagopalan have obtained many significiant results concerning shift operators on Banach spaces. Using a result of Holsztynski they classify isometric shift operators on C(X) for any compact Hausdorff space X into two (not necessarily disjoint) classes. If there exists an isometric shift operator T: C(X) → C(X) of type II, they show that X is necessarily separable. In case T is of type I, they exhibit a paticular infinite countable set of isolated points in X. Under the additional assumption that the linear functional Γ carrying f ∊ C(X) to Tf(p) ∊ is identically zero, they show that D is dense in X. They raise the question whether D will still be dense in X even when Γ ≠0. In this paper we give a negative answer to this question. In fact, given any integer l ≥ 1, we construct an example of an isometric shift operator T: C(X) —> C(X) of type I with X \ having exactly / elements, where is the closure of D in X.
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