 Essential spectrum of a closed operator

Fix the Fredholm convention for the essential spectrum of a closed densely defined linear operator $A$:Here Fredholm operator means closed range, finite-dimensional kernel of a linear map, and finite-dimensional cokernel. For an unbounded $A$, its domain can be equipped with the graph norm to formulate this as a bounded map into the ambient Banach space. Other conventions involving one-sided semi-Fredholm conditions or a zero index can differ, so the convention must accompany a statement about a non-self-adjoint linear operator.

For a self-adjoint operator on a Hilbert space, this essential spectrum is the spectrum after removing isolated eigenvalues of finite multiplicity. For the compact diagonal linear operator $A{e}_n=e_n/n$ on ${\ell }^2$, every $1/n$ is an isolated simple eigenvalue, but zero belongs to the essential spectrum. The range at zero is dense and not closed, so $A$ is not a Fredholm operator. This distinguishes an accumulation of spectral values from a single finite-dimensional eigenspace.