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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