Editing Spectral theory (section)

==A definition of spectrum==
{{Main|Spectrum (functional analysis)}}
Consider a [[Bounded linear operator|bounded linear transformation]] ''T'' defined everywhere over a general [[Banach space]]. We form the transformation:
<math display="block"> R_{\zeta} = \left( \zeta I -  T \right)^{-1}.</math>

Here ''I'' is the [[identity operator]] and ζ is a [[complex number]]. The ''inverse'' of an operator ''T'', that is ''T''<sup>−1</sup>, is defined by:
<math display="block">T T^{-1} = T^{-1} T = I. </math>

If the inverse exists, ''T'' is called ''regular''. If it does not exist, ''T'' is called ''singular''.

With these definitions, the ''[[resolvent set]]'' of ''T'' is the set of all complex numbers ζ such that ''R<sub>ζ</sub>'' exists and is [[Bounded operator|bounded]]. This set often is denoted as ''ρ''(''T''). The ''spectrum'' of ''T'' is the set of all complex numbers ζ such that ''R<sub>ζ</sub>'' <u>fails</u> to exist or is unbounded. Often the spectrum of ''T'' is denoted by ''σ''(''T''). The function ''R<sub>ζ</sub>'' for all ζ in ''ρ''(''T'') (that is, wherever ''R<sub>ζ</sub>'' exists as a bounded operator) is called the [[Resolvent formalism|resolvent]] of ''T''. The ''spectrum'' of ''T'' is therefore the complement of the ''resolvent set'' of ''T'' in the complex plane.<ref name=Lorch>{{Cite book |title=Spectral Theory |author=Edgar Raymond Lorch |year=2003 |edition=Reprint of Oxford 1962 |page=89 |publisher=Textbook Publishers |isbn=0-7581-7156-0 |url=https://books.google.com/books?id=X3U2AAAACAAJ}}</ref> Every [[eigenvalue]] of ''T'' belongs to ''σ''(''T''), but ''σ''(''T'') may contain non-eigenvalues.<ref name= Young2>{{cite book |title=''op. cit'' |author= Nicholas Young |date= 1988-07-21 |url=https://books.google.com/books?id=_igwFHKwcyYC&pg=PA81 |page=81 |publisher= Cambridge University Press |isbn=0-521-33717-8}}</ref>

This definition applies to a Banach space, but of course other types of space exist as well; for example, [[topological vector spaces]] include Banach spaces, but can be more general.<ref name=Wolff>{{Cite book |title=Topological vector spaces |author1=Helmut H. Schaefer| author2= Manfred P. H. Wolff |url=https://books.google.com/books?id=9kXY742pABoC&pg=PA36 |page=36 |year=1999 |isbn=0-387-98726-6 |edition=2nd |publisher=Springer}}</ref><ref name= Zhelobenko>{{Cite book |title=Principal structures and methods of representation theory |author=Dmitriĭ Petrovich Zhelobenko |url=https://books.google.com/books?id=3TkmvZktjp8C&pg=PA24 |isbn=	0821837311 |publisher=American Mathematical Society |year=2006}}</ref> On the other hand, Banach spaces include [[Hilbert space]]s, and it is these spaces that find the greatest application and the richest theoretical results.<ref name=Lorch2>{{Cite book |title=Spectral Theory |author=Edgar Raymond Lorch |year=2003 |isbn=0-7581-7156-0 |url=https://books.google.com/books?id=X3U2AAAACAAJ |page=57 |chapter=Chapter III: Hilbert Space|publisher=Textbook Publishers }}</ref> With suitable restrictions, much can be said about the structure of the [[Hilbert space#Spectral theory|spectra of transformations]] in a Hilbert space. In particular, for [[self-adjoint operator]]s, the spectrum lies on the [[real line]] and (in general) is a [[Decomposition of spectrum (functional analysis)|spectral combination]] of a point spectrum of discrete [[Eigenvalues#Computation of eigenvalues.2C and the characteristic equation|eigenvalues]] and a [[continuous spectrum]].<ref name=Lorch3>{{Cite book |title=Spectral Theory |author=Edgar Raymond Lorch |year=2003 |isbn=0-7581-7156-0 |url=https://books.google.com/books?id=X3U2AAAACAAJ |page=106 ''ff'' |chapter=Chapter V: The Structure of Self-Adjoint Transformations|publisher=Textbook Publishers }}</ref>