Spectrum of an element in a Banach algebra

Let $A$ be a complex unital Banach algebra with identity $1$ , and let $a\in A$ . The spectrum of $a$ in $A$ is

\sigma_A(a)=\{\lambda\in\mathbb C:\lambda 1-a \text{ is not invertible in }A\}.

Here invertibility means being a Banach-algebra invertible element. Its complement $\rho_A(a)=\mathbb C\setminus\sigma_A(a)$ is the resolvent set of $a$ . The ambient algebra is part of the notation because invertibility, and therefore the spectrum, can change when $a$ is viewed in a larger algebra. For a nonunital Banach algebra, the spectrum is defined in its unitization, with the same formula.

Compactness and spectral radius

The spectrum is a nonempty compact subset of $\mathbb C$ . Its maximum modulus is the spectral radius

r(a)=\max_{\lambda\in\sigma_A(a)}|\lambda| =\lim_{n\to\infty}\|a^n\|^{1/n} =\inf_{n\geq 1}\|a^n\|^{1/n}.

The nonemptiness assertion uses completeness and complex scalars; it can fail for real Banach algebras unless one passes to a complexification. The displayed limit is the Banach-algebra spectral-radius formula.

The

C^*

-algebra case

If $A$ is a unital $C^*$ -algebra and $a$ is self-adjoint, then $\sigma_A(a)\subseteq\mathbb R$ . If $a$ is normal, then $r(a)=\|a\|$ ; this equality need not hold for an arbitrary element. Moreover, the continuous functional calculus identifies the unital $C^*$ -algebra generated by a normal $a$ with the continuous functions on $\sigma_A(a)$ .

Ambient-algebra conventions

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. DOI record. Relevant: §1.2 on spectra, resolvents, and the spectral-radius formula.
F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, 1973. Publisher record. Relevant: spectral theory in complex unital Banach algebras.