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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?

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    $\begingroup$ Don't forget spectra in homotopy theory! $\endgroup$ – Daniel Moskovich Dec 16 '09 at 23:38
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    $\begingroup$ I'm also wondering what spectral sequences have to do with anything else spectral. $\endgroup$ – Saul Glasman Dec 17 '09 at 10:46
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Hilbert, in fact, got the term from Wilhelm Wirtinger (the first one to propose it according to, say http://www.mathphysics.com/opthy/OpHistory.html)

the paper of Wirtinger is "Beiträge zu Riemann’s Integrationsmethode für hyperbolische Differentialgleichungen, und deren Anwendungen auf Schwingungsprobleme" (1897).

In http://jeff560.tripod.com/s.html it says

"...Wirtinger drew upon the similarity with the optical spectra of molecules when he used the term "Bandenspectrum" with reference to Hill’s (differential) equation."

I haven't read Wirtinger's paper, nor do I know how reliable these sources are :)

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    $\begingroup$ Wirtinger scored a direct hit with that analogy! $\endgroup$ – Greg Kuperberg Dec 16 '09 at 21:29
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I don't know the full story, but I found the following interesting tidbits in History of Functional Analysis by Dieudonné.

On page 171 he writes the following about physicists in the 1920s: "It finally dawned upon them that their "observables" had properties which made them look like hermitian operators in Hilbert space, and that, by an extraordinary coincidence, the "spectrum" of Hilbert (a name which he had apparently chosen from a superficial analogy) was to be the central conception in the explanation of the "spectra" of atoms."

Dieudonné earlier writes (page 150): "Although Hilbert does not mention Wirtinger's paper, it is probable that he had read it (it is quoted by several of his pupils), and it may be that the name "Spectrum" which he used came from it; but it is a far cry from the vague ideas of Wirtinger to the extremely general and precise results of Hilbert."

He's referring to the same paper by Wirtinger referred to in Gjergji Zaimi's answer.

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Here is a url for "Beiträge zu Riemann’s Integrationsmethode für hyperbolische Differentialgleichungen, und deren Anwendungen auf Schwingungsprobleme"

by Wilhelm Wirtinger

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