# Decomposition of spectrum (functional analysis) # Last edited on 2013-02-05 18:10:16 by stolfilocal The [[spectrum (functional analysis)|spectrum]] of a [[linear operator]] T that operates on a [[Banach space]] X (a fundamental concept of [[functional analysis]]) consists of all [[scalar (mathematics)|scalars]] \lambda such that the operator T-\lambda does not have a bounded [[inverse function|inverse]] on X. The spectrum has a standard '''decomposition''' into three parts: * a '''point spectrum''', consisting of the [[eigenvalues and eigenvectors|eigenvalues]] of T * a '''continuous spectrum''', consisting of the scalars that are not eigenvalues but make the range of T-\lambda a [[dense subset]] of the space; * a '''residual spectrum''', consisting of all other scalars in the spectrum This decomposition is relevant to the study of [[differential equation]]s, and has applications to many branches of science and engineering. A well-known example from [[quantum mechanics]] is the explanation for the [[discrete spectrum (physics)|discrete spectral lines]] and the continuous band in the light emitted by [[excited state|excited]] atoms of [[hydrogen]]. == Definition == Let ''X'' be a [[Banach space]], ''L''(''X'') the family of [[bounded operator]]s on ''X'', and ''T'' ∈ ''L''(''X''). By [[spectrum (functional analysis)|definition]], a complex number ''λ'' is in the '''spectrum''' of ''T'', denoted ''σ''(''T''), if ''T'' − ''λ'' does not have an inverse in ''L''(''X''). If ''T'' − ''λ'' is [[injective|one-to-one]] and [[surjective|onto]], then its inverse is bounded; this follows directly from the [[open mapping theorem (functional analysis)|open mapping theorem]] of functional analysis. So, ''λ'' is in the spectrum of ''T'' if and only if ''T'' − ''λ'' is either not one-to-one or not onto. One distinguishes three separate cases: #''T'' − ''λ'' is not [[injective]]. That is, there exist two elements ''x'',''y'' in ''X'' such that (''T'' − ''λ'')(''x'') = (''T'' − ''λ'')(''y''). Then ''z'' = ''x'' − ''y'' is a non-zero vector such that ''T''(''z'') = ''λz''. In other words, ''λ'' is an eigenvalue of ''T'' in the sense of [[linear algebra]]. In this case, ''λ'' is said to be in the '''point spectrum''' of ''T'', denoted ''σ''p(''T''). #''T'' − ''λ'' is injective, and its [[range (mathematics)|range]] is a [[dense subset]] '' R'' of ''X''; but is not the whole of ''X''. In other words, there exists some element ''x'' in ''X'' such that (''T'' − ''λ'')(''y'') can be as close to ''x'' as desired, with ''y'' in ''X''; but is never equal to ''x''. It can be proved that, in this case, ''T'' − ''λ'' is not bounded below (i.e. it sends far apart elements of ''X'' too close together). Equivalently, the inverse linear operator (''T'' − ''λ'')−1, which is defined on the dense subset ''R'', is not a bounded operator, and therefore cannot be extended to the whole of ''X''. Then ''λ'' is said to be in the '''continuous spectrum''', ''σc''(''T''), of ''T''. #''T'' − ''λ'' is injective but does not have dense range. That is, there is some element ''x'' in ''X'' and a neighborhood ''N' 'of ''x'' such that (''T'' − ''λ'')(''y'') is never in ''N''. In this case, the map (''T'' − ''λ'') ''x'' → ''x'' may be bounded or unbounded, but in any case does not admit a unique extension to a bounded linear map on all of ''X''. Then ''λ'' is said to be in the '''residual spectrum''' of ''T'', ''σr''(''T''). So ''σ''(''T'') is the disjoint union of these three sets, :\sigma(T) = \sigma_p (T) \cup \sigma_c (T) \cup \sigma_r (T).