The use of the term "spectrum" to denote the prime ideals of a ring originates from the case that the ring is, say, $\mathbb{C}[T]$ where $T$ is a linear operator on a finite-dimensional vector space; then the prime spectrum (which is equal to the maximal spectrum) is precisely the set of eigenvalues of $T$.  The use of the term "spectrum" in the operator sense, in turn, seems to have originated with Hilbert, and was apparently **not** inspired by the connection to atomic spectra.  (This appears to have been a coincidence.)

A cursory Google search indicates that Hilbert may have been inspired by the significance of the eigenvalues of Laplacians, but I don't understand what this has to do with non-mathematical uses of the word "spectrum."  Does anyone know the full story here?