Banach algebras and spectrum

A complex Banach algebra A is a complete complex normed vector space with associative bilinear multiplication satisfying the product norm bound. Multiplication need not commute. If a unit is present, it acts as an identity on both sides. Completeness turns an absolutely norm-convergent formal series into an actual algebra element. Submultiplicativity then lets products pass through its partial-sum limit, providing the analytical step behind identities that initially look purely algebraic.

For an element x of norm below one, the geometric series of its nonnegative powers converges. Multiplying a finite partial sum by one minus x leaves the identity minus the next power, whose norm tends to zero. Thus the limit is a two-sided inverse. This argument works in a noncommutative algebra because all powers involved come from the same element. It also proves invertibility is stable under sufficiently small perturbations of an invertible element.

The spectrum of a consists of complex numbers lambda for which lambda times the identity minus a is not invertible inside the declared algebra. Its complement is the resolvent set. If the modulus of lambda exceeds the norm of a, factor out lambda and apply the geometric-series inverse. This places the spectrum in a closed bounded disk. A deeper complex-analytic argument shows that in a nonzero complex unital Banach algebra the spectrum is nonempty and compact; this theorem is stated rather than reproved here.

In finite-dimensional matrix algebras this recovers the eigenvalues, because noninvertibility is equivalent to zero determinant. In infinite dimensions there may be spectral values that are not eigenvalues, so the invertibility definition is more robust. The ambient algebra is also part of the definition: an inverse must belong to it. A nonunital algebra is handled by adjoining an identity and taking the spectrum in its unitization; silently using a nonexistent identity would make the formula meaningless.

The spectral-radius formula identifies the largest spectral modulus with the limit of the nth root of the norm of the nth power. In particular, spectral radius is at most norm, but equality need not hold. A nonzero nilpotent matrix has only zero in its spectrum and eventually has zero powers, while its operator norm is positive. This is one reason transient amplification cannot be read from eigenvalues alone in a nonnormal system.

The complex-field hypothesis has a simple diagnostic example. A real ninety-degree rotation matrix has determinant lambda squared plus one after subtracting a real scalar, so it has no real spectral value when considered over the real field. After complexification, its spectrum consists of i and minus i. Later C*-algebra structure restores equality of norm and spectral radius for normal elements, but that conclusion uses the adjoint and is not a generic property of Banach algebras.