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Proceedings of the American Mathematical Society

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

 

 

On normal derivations


Author: Joel Anderson
Journal: Proc. Amer. Math. Soc. 38 (1973), 135-140
MSC: Primary 47B47; Secondary 47D99
MathSciNet review: 0312313
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Abstract: Let $ {\Delta _T}$ be the derivation on $ \mathfrak{B}(\mathcal{H})$ defined by $ {\Delta _T}(X) = TX - XT(T,X \in \mathfrak{B}(\mathcal{H}))$. We prove that if $ T$ is an isometry or a normal operator, then the range of $ {\Delta _T}$ is orthogonal to the null space of $ {\Delta _T}$. Also, we prove that if $ T$ is normal with an infinite number of points in its spectrum then the closed linear span of the range and the null space of $ {\Delta _T}$ is not all of $ \mathfrak{B}(\mathcal{H})$.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0312313-6
Keywords: Derivation, commutant of a normal operator, commutant of an isometry, orthogonality, complemented subspace
Article copyright: © Copyright 1973 American Mathematical Society