Essential spectrum of a closed operator

Several essential spectra distinguish failure of upper and lower semi-Fredholm properties. For $T=A-zI$, these failures are detected at zero by the positive self-adjoint operators $T^{\ast }T$ and $T{T}^{\ast }$.

For a bounded linear operator on an infinite-dimensional complex Hilbert space, define the essential spectrum by failure of the Fredholm property for $T-\lambda I$. The Atkinson theorem identifies it with the spectrum in the Calkin algebra, hence it is compact and nonempty. On a finite-dimensional Hilbert space it is empty. For a bounded real diagonal operator on a separable infinite-dimensional space it equals the subsequential cluster set of the diagonal entries, including values of infinite multiplicity.

Let $A$ be a closed operator and let $V:D(A)\to H$ be compact when the domain carries its graph norm. If $A+V$ is closed on the same domain, compact perturbation invariance of Fredholm operators applied to the maps $D(A)\to H$ proves invariance of the Fredholm essential spectrum. For self-adjoint operators this is also the essential spectrum obtained by deleting isolated finite-multiplicity eigenvalues. A bounded potential tending to zero at spatial infinity is relatively compact with respect to a second-order constant-coefficient operator: local Rellich-Kondrachov compactness theorem gives compactness on bounded intervals and the small tails give compactness globally.

For bounded $L$ and compact $K$, use the Fredholm convention ${\Sigma }_{\mathrm{e}}(L)=\{z:L-zI\text{ is not Fredholm}\}$. Applying compact perturbation invariance of Fredholm operators to every shift proves the equality. This general theorem needs no self-adjointness; other definitions of essential spectrum for nonnormal operators must be distinguished.

For a bounded self-adjoint operator $L$ on a complex Hilbert space, defineThis equals the spectrum of a bounded operator after removing its isolated eigenvalues of finite multiplicity. To see isolation when the kernel is finite and the shifted range is closed, restrict the shifted operator to its kernel complement: it is invertible there by image-kernel orthogonality for an adjoint. A Neumann-series perturbation leaves nearby nonzero shifts invertible on that complement and on the finite kernel. The closed range must concern the shifted operator, not the original one.

If $L$ is bounded and self-adjoint on a complex Hilbert space and $K$ is compact and self-adjoint, thenA singular Weyl sequence for $L$ stays singular for $L+K$, because compact operators send weak convergence to norm convergence and therefore $K{f}_n\to 0$. Applying the same argument with $-K$ proves the reverse inclusion. Finite-multiplicity isolated eigenvalues can move under such perturbations; the essential spectrum of a bounded self-adjoint operator is unchanged.

The discrete spectrum consists of isolated eigenvalues of finite algebraic multiplicity.