Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 8 / 5 / b

For the Fredholm convention, defineThe equality follows from the Atkinson theorem. The quotient is a unital complex Banach algebra when $H$ is infinite-dimensional. Its invertible set is open by the Neumann series, so the spectrum is closed, and it lies in the disk $|\lambda |\le \Vert [T]\Vert$.

For nonemptiness, suppose the quotient resolvent $R(\lambda )=(\lambda 1-[T]{)}^{-1}$ existed for every complex $\lambda$. It is analytic everywhere, and outside a sufficiently large disk its Neumann expansion gives $\Vert R(\lambda )\Vert \le 1/(|\lambda |-\Vert [T]\Vert )$. Continuity bounds it on the remaining compact disk. Every continuous linear functional applied to $R$ is therefore a bounded entire scalar function, constant by Liouville's theorem and zero because it tends to zero at infinity. The Hahn-Banach theorem, or separation by the dual, forces $R(\lambda )=0$, contradicting its inverse identity. The essential spectrum is nonempty and closed.

Infinite dimension is necessary: every operator on a finite-dimensional Hilbert space is Fredholm, so its essential spectrum in this convention is empty. The proof works on nonseparable infinite-dimensional spaces as well; it needs neither self-adjointness nor a sequence basis.