Exercise 17, § 2

Tag 0274, permanent. Cite it as [Bourbaki, Tag 0274].

Spectral Theories, I, p. 158

Let $A$ be a normed algebra. For every $x\in A$ , set:

$\lambda (x) =$ inf $_{y\not=0}\frac{\|xy\|}{\|y\|}\lambda '(x) =$ inf $_{y\not=0}\frac{\|yx\|}{\|y\|}$ .

a) One has

|\lambda (x)-\lambda (y)|\leqslant \|x-y\|,|\lambda '(x)-\lambda '(y)|\leqslant \|x-y\|

\lambda (x)\lambda (y)\leqslant \lambda (xy)\leqslant \|x\|\lambda (y),\lambda '(x)\lambda '(y)\leqslant \lambda '(xy)\leqslant \lambda '(x)\|y\|

b) The set of topological zero divisors on the left (resp right) in $A$ is closed.

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