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A Matricial Identity Involving the Self-Commutator of a Commuting n-Tuple
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A Matricial Identity Involving the Self-Commutator of a Commuting n-Tuple

Raul E Curto and Renyi Jian
Proceedings of the American Mathematical Society, Vol.121(2), pp.461-464
06/1994
DOI: 10.2307/2160422

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Abstract

For a commuting n-tuple a = (a1, ..., an) of elements of a unital C*-algebra A, we establish a matricial identity linking the self-commutator of a to the 2n - 1 × 2n - 1 matrix â 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, σT(a) = σr(a).

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