In the book, a first course in functional analysis by D.Somasundaram, it is mentioned that $\sigma(x) = \{z \in\mathbb{C}: \vert z\vert\leq\Vert x \Vert\}$ But the proof is given only for one side inclusion..that is., The spectrum of an element x in a Complex Banach Algebra is a subset of the closed disk with centre $0$ and radius $\Vert x\Vert$. Is the reverse inclusion true? So that they become equal. (I cannot find the proof of reverse inclusion in any book so far I have seen)
2026-03-25 22:05:29.1774476329
Is the spectrum of an element in an algebra, $\sigma(x) = \{z \in\mathbb{C}: \vert z\vert\leq\Vert x \Vert\}$?
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The spectrum of an element x denoted by sigma(x)={z : (x-ze)is singular} is a closed and bounded subset of the complex plane and it is only a subset of the closed disk of radius ||x|| and not equal to the disk of radius ||x||. It may be either a notational complexity of the book or a misprint that the spectrum of an element x in an algebra equals the closed disk of radius||x||.