The spectrum of the Cesàro operator on c0(c0)
Loading...
Date
1989-01
Authors
Okutoyi, J. I.
Thorpe, B.
Journal Title
Journal ISSN
Volume Title
Publisher
Cambridge University Press
Abstract
In a recent paper [6], the spectrum of the Cesàro operator C on c0 (the space of null sequences of complex numbers with the sup norm) was obtained by finding the eigenvalues of the adjoinoperator on and showing that the operator (C–λI)−1 lies in B(c0) for all λ outside the closure of this set of eigenvalues. In this paper we apply a similar method to find the spectrum of the two-dimensional Cesàro operator on a space of double sequences c0(c0) (defined in §2). We shall introduce a simplification to the proof in [6] by observing that (C – λI)−1, when it exists, is a Hausdorff summability method (see page 288 of [11] for the single variable case on the space of convergent sequences c), and the crux of our proof is to show that the moment constant associated with the method (C – λI)−1 is regular for the space c0(c0) and the set of λ under consideration. It turns out that c0(c0) c0 c0 (see page 237 of [7]) and that the two-dimensional Cesàro operator on c0(c0) is the tensor product C C of the Cesàro operator C on c0. Thus our result gives a direct proof that the spectrum σ(C C) equals σ(C)σ(C), which is a special case of the result of Schechter in [8].
Description
DOI: http://dx.doi.org/10.1017/S0305004100001420
Keywords
Citation
Mathematical Proceedings of the Cambridge Philosophical Society / Volume 105 / Issue 01 / January 1989, pp 123-129