The spectrum of the Cesàro operator on c0(c0)
Okutoyi, J. I.
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In a recent paper , 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  by observing that (C – λI)−1, when it exists, is a Hausdorff summability method (see page 288 of  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 ) 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 .