spectrum
In finite dimensions, a matrix is captured by its eigenvalues — the special scaling factors along which it merely stretches. The spectrum is the right generalization of that list of eigenvalues to infinite-dimensional spaces, where an operator can fail to be invertible for reasons subtler than having a nonzero null vector. Think of it as the complete set of scalars at which the operator becomes 'singular' in some way.
Let T be a bounded linear operator on a complex Banach space X. The spectrum sigma(T) is the set of complex numbers lambda such that the operator T - lambda*I has no bounded inverse on X, where I is the identity. Its complement, the set where T - lambda*I IS boundedly invertible, is the resolvent set. For a bounded operator on a nonzero complex Banach space the spectrum is always a nonempty compact subset of the complex plane, contained in the closed disk of radius equal to the operator norm.
A frequent mistake is to equate the spectrum with the eigenvalues. Every eigenvalue lies in the spectrum, but in infinite dimensions the spectrum can contain points that are not eigenvalues at all: T - lambda*I may be injective yet fail to be surjective or fail to have bounded inverse. The spectrum splits into the point spectrum (genuine eigenvalues), the continuous spectrum, and the residual spectrum, each describing a different way invertibility breaks down.
Let S be the right shift on l^2: S(x_1, x_2, ...) = (0, x_1, x_2, ...). It has NO eigenvalues at all, yet its spectrum is the entire closed unit disk |lambda| <= 1. So sigma(S) is huge while the point spectrum is empty.
Spectrum without any eigenvalue.
Over the real field a bounded operator can have empty spectrum (rotation by 90 degrees in R^2 has no real eigenvalues), so the nonemptiness theorem genuinely needs complex scalars.