Integral of secant cubed

In operator theory, Atkinson's theorem (named for Frederick Valentine Atkinson) gives a characterization of Fredholm operators.

The theorem

Let H be a Hilbert space and L(H) the set of bounded operators on H. The following is the classical definition of a Fredholm operator: an operator T ∈ L(H) is said to be a Fredholm operator if the kernel Ker(T) is finite dimensional, Ker(T*) is finite dimensional (where T* denotes the adjoint of T), and the range 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.

Sketch of proof

The outline of a proof is as follows. For the ⇒ implication, express H as the orthogonal direct sum

$H={\begin{matrix}\operatorname {Ker} (T)^{\perp }\\\oplus \\\operatorname {Ker} (T).\end{matrix}}$

The restriction T : Ker(T)^⊥ → Ran(T) is a bijection, and therefore invertible by the open mapping theorem. 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 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 + C₂ for some compact operator C₂. If x ∈ Ker(T), then STx = x + C₂x = 0. So Ker(T) is contained in an eigenspace of C₂, which is finite dimensional (see 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: let F be a finite-rank operator such that ||F − C₂|| < r. Then for every x in Ker(F),

||S||·||Tx|| ≥ ||STx|| = ||x + C₂x|| = ||x + Fx +C₂x − Fx|| ≥ ||x|| − ||C₂ − 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.

References

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