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Fredholm Alternative


For any nonzero lambda in C, either

1. The equation Tv-lambdav=0 has a nonzero solution v, or

2. The equation Tv-lambdav=f has a unique solution v for any function f.

In the second case, the solution v depends continuously on f. The Fredholm alternative applies when T:V->V is a compact operator, such as an integral operator with a smooth integral kernel.

The Fredholm alternative can be restated as follows: any nonzero lambda which is not an eigenvalue of a compact operator is in the resolvent, i.e., (T-lambdaI)^(-1), is bounded. The basic special case is when V is finite-dimensional, in which case any nondegenerate matrix is diagonalizable.


See also

Compact Operator, Eigenvalue, Fredholm Operator, Integral Kernel, Nonsingular Matrix, Spectral Theory

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Fredholm Alternative." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/FredholmAlternative.html

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