Isolated Points of the Spectra of Conservative Matrices

We show that the only possible isolated points in the spectrum of a conservative triangular matrix are its diagonal elements, and that any Hausdorff method corresponding to an absolutely continuous mass function is in the norm closure of the analytic methods. Let oj represent the space of all real and complex sequences and let c (cQ or m) represent the subspace of all convergent sequences (null sequences or bounded sequences). Let B(c) represent the complex Banach algebra of bounded operators on c with the usual norm topology. Let A = (a , ) be any infinite matrix and let, for x in a>, Ax denote the transformed sequence, provided it exists. Let cA = \x £ a>: AX e c\. A is called conservative (coercive) if cA D c (cA 2 m). For the properties of conservative (coercive) matrices and Hausdorff methods the reader might refer to [2J. A matrix A is called triangular if a , = 0 whenever k > n and a triangular matrix A is called a triangle if a ^0 for any n. A denotes the Banach algebra of triangular matrices in B(c). If A e B(c) then the set of complex numbers z such that A — zl is not invertible in B(c) is called the spectrum of A and is denoted by a(A). That a(A) is nonempty and compact is well known [6, p. 418]. In § 1 we show that if A is in A then the only possible isolated points in a(A) ate the a 's. This answers Question 2 of [9] negatively. We also show that the spectrum of a multiplicative Hausdorff method is always connected. In § 2 we show that the Hausdorff methods corresponding to absolutely continuous mass functions are indeed in the norm closure of the analytic methods of [3j. We then give an application of this result and end the paper with an open question. Presented to the Society November 17, 1972; received by the editors August 8, 1973. AMS (MOS) subject classifications (1970). Primary 40H05, 40G05; Secondary 47B99. Copyright © 1975, American Mathematical Society 74 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use SPECTRA OF CONSERVATIVE MATRICES 75