Corollary 3

Let A be a unital Banach algebra and $x$ an invertible element of A such that $\|x\|=\|x^{-1}\|= 1$. Then Sp($x$)$\subset \mathbf{U}$.

Let Δ be the disc with center 0 and radius 1 in $\mathbf{C}$. By theorem 1 b) and the fact that $\varrho (x)\leqslant \|x\|$, one has Sp($x$)$\subset \Delta$. Analogously, Sp($x$)$^{-1}$ = Sp($x^{-1}$)$\subset \Delta$, whence the corollary (cf. I, p. 2, remark 4).