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A spectral mapping theorem relates the spectrum of an operator to the spectrum after applying a function to it. For a bounded linear operator $T$ on a complex Banach space and a polynomial $p$,Factor $p(z)-\mu$ over the complex numbers. If no root lies in $\sigma (T)$, every factor $T-\lambda I$ is invertible, so their product is invertible. Conversely a product of commuting factors can be invertible only if each factor is invertible, giving the reverse inclusion.

For a matrix with eigenvalues one and three, $T^2+I$ has eigenvalues two and ten. More general functions require a specified functional calculus; an unrestricted substitution rule for unbounded operators is not automatic.

Vahid Shirbisheh authored lectures on C-star algebras connecting Banach algebras, spectral theory and functional calculus. The notes develop the Gelfand transform and the correspondence between commutative C-star algebras and appropriate topological spaces. This explains how an abstract algebra can encode continuous functions and recover geometric information from its multiplicative structure.^[ref]