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Proceedings of the American Mathematical Society

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

 

 

A matricial identity involving the self-commutator of a commuting $ n$-tuple


Authors: Raúl E. Curto and Ren Yi Jian
Journal: Proc. Amer. Math. Soc. 121 (1994), 461-464
MSC: Primary 47B47; Secondary 46L99, 47A13, 47B20
DOI: https://doi.org/10.1090/S0002-9939-1994-1182700-9
MathSciNet review: 1182700
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Abstract: For a commuting n-tuple $ a = ({a_1}, \ldots ,{a_n})$ of elements of a unital $ {C^ \ast }$-algebra $ \mathcal{A}$, we establish a matricial identity linking the self-commutator of a to the $ {2^{n - 1}} \times {2^{n - 1}}$ matrix $ \hat a$ that detects the Taylor invertibility of a. As a consequence, we obtain a simple proof of a result of D. Xia (Oper. Theory: Adv. Appl. 48 (1990), 423-448), which states that for commuting t-hyponormal n-tuples, $ {\sigma _T}(a) = {\sigma _r}(a)$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1182700-9
Keywords: Self-commutator, $ {C^ \ast }$-algebra, Taylor spectrum, right spectrum, t-hyponormal
Article copyright: © Copyright 1994 American Mathematical Society