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Proceedings of the American Mathematical Society

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



Analytic functions, ideals, and derivation ranges

Author: R. E. Weber
Journal: Proc. Amer. Math. Soc. 40 (1973), 492-496
MSC: Primary 47A60
MathSciNet review: 0353025
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Abstract: When $ A$ is in the Banach algebra $ \mathcal{B}(\mathcal{H})$ of all bounded linear operators on a Hilbert space $ \mathcal{H}$, the derivation generated by $ A$ is the bounded operator $ {\Delta _A}$ on $ \mathcal{B}(\mathcal{H})$ defined by $ {\Delta _A}(X) = AX - XA$. It is shown that (i) if $ B$ is an analytic function of $ A$, then the range of $ {\Delta _B}$ is contained in the range of $ {\Delta _A}$; (ii) if $ U$ is a nonunitary isometry, then the range of $ {\Delta _U}$, contains nonzero left ideals; (iii) if $ U$ and $ V$ are isometries with orthogonally complemented ranges, then the span of the ranges of the corresponding derivations is all of $ \mathcal{B}(\mathcal{H})$.

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Keywords: Derivation ranges, left ideals, analytic functions, orthogonally complemented ranges
Article copyright: © Copyright 1973 American Mathematical Society