On weak* continuous operators on ℬ(ℋ)

Weber, Robert E.

doi:10.1090/S0002-9939-1981-0630046-8

On weak$^{\ast }$ continuous operators on $\mathcal {B}(\mathcal {H})$
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by Robert E. Weber

Proc. Amer. Math. Soc. 83 (1981), 735-742

DOI: https://doi.org/10.1090/S0002-9939-1981-0630046-8

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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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