Spectrum of an element

Summary

The spectrum of an element $a$ of a unital algebra is the set of scalars $\lambda$ for which $a - \lambda 1$ has no inverse. For square matrices it is the set of eigenvalues, and for bounded operators it can be larger than the set of eigenvalues. In a complex Banach algebra the spectrum is a nonempty compact set. It is the starting point of spectral theory and of the functional calculus.