A simple proof of the Fredholm alternative and a characterization of the Fredholm operators

Details A simple proof of the Fredholm alternative and a characterization of the Fredholm operators A simple proof of the Fredholm alternative and a characterization of the Fredholm operators

2000-11-17

arxiv.org/abs/math/0011133v1

Mathematics

Functional Analysis

6pp, a selfcontained, short and simple proof of the Freholm alternative and of a characterization of Fredholm operators. The p

Scientific paper

Let $A$ be a linear bounded operator in a Hilbert space $H$, $N(A)$ and $R(A)$ its null-space and range, and $A^*$ its adjoint. The operator $A$ is called Fredholm iff $dim N(A)= dim N(A^*):=n<\infty$ and $R(A)$ and $R(A^*)$ are closed subspaces of $H$. A simple and short proof is given of the following known result: $A$ is Fredholm iff $A=B+F$, where $B$ is an isomorphism and $F$ is a finite-rank operator. The proof consists in reduction to a finite-dimensional linear algebraic system which is equivalent to the equation $Au=f$ in the case of Fredholm operators.

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