- Fredholm alternative
In
mathematics , the Fredholm alternative, name afterIvar Fredholm , is one ofFredholm's theorem s and is a result inFredholm theory . It may be expressed in several ways, as a theorem oflinear algebra , a theorem ofintegral equation s, or as a theorem onFredholm operator s. Part of the result states that, a non-zero complex number in the spectrum of acompact operator is an eigenvalue.Linear algebra
If "V" is an "n"-dimensional
vector space and T:V o V is alinear transformation , then exactly one of the following holds:#For each vector "v" in "V" there is a vector "u" in "V" so that T(u) = v. In other words: T is surjective.
#dim(ker(T)) > 0.Integral equations
Let K(x,y) be an
integral kernel , and consider thehomogeneous equation , theFredholm integral equation ,:lambda phi(x)- int_a^b K(x,y) phi(y) ,dy = 0
and the inhomogeneous equation
:lambda phi(x) - int_a^b K(x,y) phi(y) ,dy = f(x).
The Fredholm alternative states that, for any non-zero fixed
complex number lambda in mathbb{C}, either the first equation has a non-trivial solution, or the second equation has a solution for all f(x).A sufficient condition for this theorem to hold is for K(x,y) to be
square integrable on the rectangle a,b] imes [a,b] (where "a" and/or "b" may be minus or plus infinity).Functional analysis
Results on the
Fredholm operator generalize these results to vector spaces of infinite dimensions,Banach space s.Correspondence
Loosely speaking, the correspondence between the linear algebra version, and the integral equation version, is as follows: Let
:T=lambda - K
or, in index notation,
:T(x,y)=lambda delta(x-y) - K(x,y)
with delta(x-y) the
Dirac delta function . Here, "T" can be seen to be anlinear operator acting on a Banach space "V" of functions phi(x), so that:T:V o V
is given by
:phi mapsto psi
with psi given by
:psi(x)=int_a^b T(x,y) phi(y) ,dy
In this language, the integral equation alternatives are seen to correspond to the linear algebra alternatives.
Alternative
In more precise terms, the Fredholm alternative only applies when "K" is a
compact operator . From Fredholm theory, smooth integral kernels are compact operators. The Fredholm alternative may be restated in the following form: a nonzero lambda is either aneigenvalue of "K", or it lies in the domain of theresolvent :R(lambda; K)= (K-lambda operatorname{Id})^{-1}.
See also
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Spectral theory of compact operators References
* E.I. Fredholm, "Sur une classe d'equations fonctionnelles", "Acta Math." , 27 (1903) pp. 365–390.
* A. G. Ramm, " [http://www.math.ksu.edu/~ramm/papers/419amm.pdf A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators] ", "American Mathematical Monthly", 108 (2001) p. 855.
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