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The spectrum of the Cesàro operator on c0(c0)

Published online by Cambridge University Press:  24 October 2008

J. Okutoyi  and
B. Thorpe
J. Okutoyi
Affiliation:
Department of Mathematics, Kenyatta University College, Nairobi, Kenya
B. Thorpe
Affiliation:
Department of Mathematics, University of Birmingham, Birmingham B15 2TT
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Extract

1. 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 adjoint operator 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) ≅ c0c0 (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].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 1 , January 1989 , pp. 123 - 129
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

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