# Atkinson's theorem > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Atkinson's_theorem > Markdown URL: https://mediated.wiki/source/Atkinson's_theorem.md > Source: https://en.wikipedia.org/wiki/Atkinson's_theorem > Source revision: 1284299769 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) In [operator theory](/source/Operator_theory), **Atkinson's theorem** (named for [Frederick Valentine Atkinson](/source/Frederick_Valentine_Atkinson)) gives a characterization of [Fredholm operators](/source/Fredholm_operator). ## The theorem Let *H* be a [Hilbert space](/source/Hilbert_space) and *L*(*H*) the set of bounded operators on *H*. The following is the classical definition of a [Fredholm operator](/source/Fredholm_operator): an operator *T* ∈ *L*(*H*) is said to be a Fredholm operator if the [kernel](/source/Kernel_(linear_operator)) Ker(*T*) is finite-dimensional, Ker(*T**) is finite-dimensional (where *T** denotes the [adjoint](/source/Hermitian_adjoint) of *T*), and the [range](/source/Range_of_a_function) 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* + *C*1 and *ST* = *I* + *C*2 for some bounded operator *S* and [compact operators](/source/Compact_operator) *C*1 and *C*2. In other words, an operator *T* ∈ *L*(*H*) is Fredholm, in the classical sense, if and only if its projection in the [Calkin algebra](/source/Calkin_algebra) is invertible. ### Sketch of proof The outline of a proof is as follows. For the ⇒ implication, express *H* as the orthogonal direct sum - H = \operatorname{Ker}(T)^\perp \oplus \operatorname{Ker} (T). The restriction *T* : Ker(*T*)⊥ → Ran(*T*) is a bijection, and therefore invertible by the [open mapping theorem](/source/Open_mapping_theorem_(functional_analysis)). Extend this inverse by 0 on Ran(*T*)⊥ = Ker(*T**) to an operator *S* defined on all of *H*. Then *I* − *TS* is the [finite-rank](/source/Finite-rank_operator) projection onto Ker(*T**), and *I* − *ST* is the projection onto Ker(*T*). This proves the only if part of the theorem. For the converse, suppose now that *ST* = *I* + *C*2 for some compact operator *C*2. If *x* ∈ Ker(*T*), then *STx* = *x* + *C*2*x* = 0. So Ker(*T*) is contained in an eigenspace of *C*2, which is finite-dimensional (see [spectral theory of compact operators](/source/Spectral_theory_of_compact_operators)). Therefore, Ker(*T*) is also finite-dimensional. The same argument shows that Ker(*T**) is also finite-dimensional. To prove that Ran(*T*) is closed, we make use of the [approximation property](/source/Approximation_property): let *F* be a [finite-rank operator](/source/Finite-rank_operator) such that ||*F* − *C*2|| < *r*. Then for every *x* in Ker(*F*), - ||*S*||⋅||*Tx*|| ≥ ||*STx*|| = ||*x* + *C*2*x*|| = ||*x* + *Fx* +*C*2*x* − *Fx*|| ≥ ||x|| − ||*C*2 − *F*||⋅||x|| ≥ (1 − *r*)||*x*||. Thus *T* is bounded below on Ker(*F*), which implies that *T*(Ker(*F*)) is closed. On the other hand, *T*(Ker(*F*)⊥) is finite-dimensional, since Ker(*F*)⊥ = Ran(*F**) is finite-dimensional. Therefore, Ran(*T*) = *T*(Ker(*F*)) + *T*(Ker(*F*)⊥) is closed, and this proves the theorem. A more complete treatment of Atkinson's Theorem is in the reference by Arveson: it shows that if B is a Banach space, an operator is Fredholm if and only if it is invertible modulo a finite rank operator (and that the latter is equivalent to being invertible modulo a compact operator, which is significant in view of Enflo's example of a separable, reflexive Banach space with compact operators that are not norm-limits of finite rank operators). For Banach spaces, a Fredholm operator is one with finite dimensional kernel and range of finite codimension (equivalent to the kernel of its adjoint being finite dimensional). Note that the hypothesis that Ran(*T*) is closed is redundant since a space of finite codimension that is also the range of a bounded operator is always closed (see Arveson reference below); this is a consequence of the open-mapping theorem (and is not true if the space is not the range of a bounded operator, for example the kernel of a discontinuous linear functional). ## References - Atkinson, F. V. (1951). "The normal solvability of linear equations in normed spaces". *Mat. Sb.*. **28** (70): 3–14. Zbl 0042.12001. - Arveson, William B., A Short Course on Spectral Theory, Springer Graduate Texts in Mathematics, vol 209, 2002, ISBN 0387953000 --- Adapted from the Wikipedia article [Atkinson's theorem](https://en.wikipedia.org/wiki/Atkinson's_theorem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Atkinson's_theorem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.