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

Curto, Raúl E.; Jian, Ren Yi

doi:10.1090/S0002-9939-1994-1182700-9

A matricial identity involving the self-commutator of a commuting $n$-tuple
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by Raúl E. Curto and Ren Yi Jian PDF

Proc. Amer. Math. Soc. 121 (1994), 461-464 Request permission

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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