Spectral radius formula for a Banach algebra

Statement

Let $A$ be a complex Banach algebra and $a\in A$ . The spectral radius formula states that the spectral radius of $a$ satisfies

r_A(a)=\lim_{n\to\infty}\lVert a^n\rVert^{1/n} =\inf_{n\geq1}\lVert a^n\rVert^{1/n}.

If $A$ is nonunital, $r_A(a)$ means the spectral radius computed in its unitization. The limit exists because $\lVert a^{m+n}\rVert\leq\lVert a^m\rVert\lVert a^n\rVert$ , so the logarithms form a subadditive sequence. The formula identifies spectral size with asymptotic power growth.

Proof idea

The inequality $r_A(a)\leq\lVert a^n\rVert^{1/n}$ follows by applying the spectral mapping theorem: $\sigma_A(a^n)=\{\lambda^n:\lambda\in\sigma_A(a)\}$ . Conversely, the resolvent expansion

(\lambda1_A-a)^{-1} =\sum_{n=0}^{\infty}\lambda^{-n-1}a^n

converges whenever $|\lambda|$ exceeds the limiting power-growth rate. Hence no spectral point can lie outside the corresponding disk.

Consequences and examples

An element is quasinilpotent exactly when $\lVert a^n\rVert^{1/n}\to0$ . For a normal element $a$ of a $C^*$ -algebra, $\lVert a^n\rVert=\lVert a\rVert^n$ , so $r_A(a)=\lVert a\rVert$ . In contrast, a nonzero nilpotent matrix has spectral radius zero although its norm is positive, showing why $r_A(a)=\lVert a\rVert$ fails for general elements.

Conventions and scope

The limit concerns the $n$ -th roots of norms, not the sequence $\lVert a^n\rVert$ itself. The equality with the infimum is an application of the subadditive lemma after taking logarithms, with zero powers handled separately. Over a real Banach algebra, the usual spectral radius is defined after complexification; the core theorem is stated directly for complex algebras.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: Theorem 1.2.3 on the spectral radius formula.
F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, 1973. DOI record. Relevant: “Concepts and Elementary Results” on spectra and the spectral radius in complete normed algebras.