High Quality Content by WIKIPEDIA articles! In operator theory, Atkinson's theorem gives a characterization of Fredholm operators.Let H be a Hilbert space and L(H) the bounded operators on H. The following is the classical definition of a Fredholm operator: a T ? L(H) is said to be a Fredholm operator if the kernel of T Ker(T) is finite dimensional, Ker(T*) is finite dimensional, and the range of T Ran(T) is closed. Atkinson's theorem states: A T ? L(H) is a Fredholm operator if and only if T is invertible modulo compact perturbation, i.e. TS = I + C1 and ST = I + C2 for some bounded operator S and compact operators C1 and C2. In other words, an operator T ? L(H) is Fredholm, in the classical sense, if and only if its projection in the Calkin algebra is invertible.

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