Fredholm alternative

In mathematics, named after Ivar Fredholm Fredholm alternative is a result of the Fredholm theory. It can be expressed in various ways: as a theorem of linear algebra, as a theorem on integral equations or as a theorem on Fredholm operators. In particular, it states that a complex number not equal to 0 in the spectrum of a compact operator is an eigenvalue.

Version of linear algebra

In an n-dimensional vector space V applies to a linear map exactly one of the following:

Fredholm integral equations

Be a core integratable. Consider the homogeneous Fredholm integral equation,

And the inhomogeneous equation

The Fredholm alternative states that, for a complex number, either the first equation has a nontrivial solution, or the second equation has a solution for arbitrary right-hand sides.

A sufficient condition for that set is considered, the Quadratintegrierbarkeit of on the rectangle ( where a and / or b may be infinite even plus or minus).

Fredholm alternative

Statement

Be a compact operator on and be with. Then a Fredholm operator with Fredholm index 0 is the Fredholm alternative is now:

• Either both the homogeneous equation and the adjoint equation only the trivial solution zero and thus the inhomogeneous equations and uniquely solvable,
• Or the homogeneous equation and the adjoint equation have exactly linearly independent solutions, and thus would be the inhomogeneous equation solvable if and only if.

In connection with the integral equations

Note that the delta function is the identity of the fold. Be a Banach space for example and be a Fredholm operator, which by

Is defined, which must apply to obtain a foreign rail operator. So is a compact operator and you can see that this statement generalizes the statement about the Fremdholm'schen integral equations.

The Fredholm alternative can then be formulated as follows: A is either an eigenvalue of, or it is in the resolvent