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Proceedings of the American Mathematical Society

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

 

 

Not every $ d$-symmetric operator is GCR


Author: C. Ray Rosentrater
Journal: Proc. Amer. Math. Soc. 81 (1981), 443-446
MSC: Primary 47B47; Secondary 46L05
DOI: https://doi.org/10.1090/S0002-9939-1981-0597659-3
MathSciNet review: 597659
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Abstract: Let $ T$ be an element of $ \mathcal{B}(\mathcal{H})$, the algebra of bounded linear operators on the Hilbert space $ \mathcal{H}$. The derivation induced by $ T$ is the map $ {\delta _T}(X) = TX - XT$ from $ \mathcal{B}(\mathcal{H})$ into itself. $ T$ is $ d$-symmetric if the norm closure of the range of $ {\delta _T}$, $ \mathcal{R}{({\delta _T})^\_}$, is closed under taking adjoints. This paper answers the question of whether every $ d$-symmetric operator is GCR by giving an example of an NGCR weighted shift that is also $ d$-symmetric.


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DOI: https://doi.org/10.1090/S0002-9939-1981-0597659-3
Article copyright: © Copyright 1981 American Mathematical Society