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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Ideals of regular operators on $ l\sp{2}$


Authors: W. Arendt and A. R. Sourour
Journal: Proc. Amer. Math. Soc. 88 (1983), 93-96
MSC: Primary 47D30; Secondary 47B55
MathSciNet review: 691284
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Abstract: Let $ {\mathcal{L}^r}$ be the Banach algebra (and Banach lattice) of all regular operators on $ {l^{^2}}$, i.e. the algebra of all operators $ A$ on $ {l^2}$ which are given by a matrix $ ({a_{mn}})$ such that $ (\left\vert {{a_{mn}}} \right\vert)$ defines a bounded operator $ \left\vert A \right\vert$. We show that there exists exactly one nontrivial closed subspace of $ {\mathcal{L}^r}$ which is both a lattice-ideal and an algebra-ideal of $ {\mathcal{L}^r}$, namely the space $ {\mathcal{K}^r} = \{ A \in {\mathcal{L}^r}:\left\vert A \right\vert{\text{ is compact}}\} $. We also show that every nontrivial ideal in $ {\mathcal{L}^r}$ is included in $ {\mathcal{K}^r}$.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0691284-3
Article copyright: © Copyright 1983 American Mathematical Society