Editing Spectral theory (section)

==Mathematical background==
The name ''spectral theory'' was introduced by [[David Hilbert]] in his original formulation of [[Hilbert space]] theory, which was cast in terms of [[quadratic form]]s in infinitely many variables. The original [[spectral theorem]] was therefore conceived as a version of the theorem on [[Principal axis theorem|principal axes]] of an [[ellipsoid]], in an infinite-dimensional setting. The later discovery in [[quantum mechanics]] that spectral theory could explain features of [[Emission spectrum|atomic spectra]] was therefore fortuitous. Hilbert himself was surprised by the unexpected application of this theory, noting that "I developed my theory of infinitely many variables from purely mathematical interests, and even called it 'spectral analysis' without any presentiment that it would later find application to the actual spectrum of physics."<ref>{{cite web|last1=Steen|first1=Lynn Arthur|title=Highlights in the History of Spectral Theory|url=http://www.stolaf.edu/people/steen/Papers/73spectral.pdf|website=St. Olaf College|access-date=14 December 2015|url-status=dead|archive-url=https://web.archive.org/web/20160304050120/http://www.stolaf.edu/people/steen/Papers/73spectral.pdf|archive-date=4 March 2016}}</ref>

There have been three main ways to formulate spectral theory, each of which find use in different domains. After Hilbert's initial formulation, the later development of abstract [[Hilbert space]]s and the spectral theory of single [[normal operator]]s on them were well suited to the requirements of [[physics]], exemplified by the work of [[John von Neumann|von Neumann]].<ref name= vonNeumann>

{{Cite book |title=The mathematical foundations of quantum mechanics; ''Volume 2 in Princeton'' Landmarks in Mathematics ''series'' |author=John von Neumann |url=https://books.google.com/books?id=JLyCo3RO4qUC&q=mathematical+foundations+of+quantum+mechanics+inauthor:von+inauthor:neumann |isbn=0-691-02893-1 |year=1996 |publisher =Princeton University Press |edition=Reprint of translation of original 1932 }}

</ref> The further theory built on this to address [[Banach algebra]]s in general. This development leads to the [[Gelfand representation]], which covers the [[commutative Banach algebra|commutative case]], and further into [[non-commutative harmonic analysis]].

The difference can be seen in making the connection with [[Fourier analysis]]. The [[Fourier transform]] on the [[real line]] is in one sense the spectral theory of [[derivative|differentiation]] as a [[differential operator]]. But for that to cover the phenomena one has already to deal with [[generalized eigenfunction]]s (for example, by means of a [[rigged Hilbert space]]). On the other hand, it is simple to construct a [[group algebra of a locally compact group|group algebra]], the spectrum of which captures the Fourier transform's basic properties, and this is carried out by means of [[Pontryagin duality]].

One can also study the spectral properties of operators on [[Banach spaces]]. For example, [[compact operator]]s on Banach spaces have many spectral properties similar to that of [[Matrix (mathematics)|matrices]].