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Proceedings of the American Mathematical Society

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On weak$ \sp{\ast} $ continuous operators on $ \mathcal{B}(\mathcal{H})$


Author: Robert E. Weber
Journal: Proc. Amer. Math. Soc. 83 (1981), 735-742
MSC: Primary 47A05; Secondary 47A10, 47D25
DOI: https://doi.org/10.1090/S0002-9939-1981-0630046-8
MathSciNet review: 630046
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Abstract: If $ \Delta $ is a weak* continuous bounded linear operator on $ \mathcal{B}(\mathcal{H})$ that fixes the ideal of compact operators $ \mathcal{K}$ and $ {\Delta _0}$ and $ \delta $ are the induced maps on $ \mathcal{K}$ and $ \mathcal{B}(\mathcal{H})/\mathcal{K}$ then it is shown that $ \Delta $ has closed range, has dense range, is bounded below, or is onto if and only if both $ {\Delta _0}$ and $ \delta $ have the same property. These results are then applied to the operator $ X \to AXB$.


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DOI: https://doi.org/10.1090/S0002-9939-1981-0630046-8
Keywords: Weak* continuous, Calkin algebra, operator range, spectrum, bounded below
Article copyright: © Copyright 1981 American Mathematical Society

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