= Essential spectrum of a closed operator
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= Essential spectrum
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= Essential spectra of a closed operator
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Fix the Fredholm convention for the <essential spectrum> of a closed densely defined <linear operator> $A$:
$$
\sigma_{\mathrm{ess}}(A)=\{\lambda\in\mathbb C:A-\lambda I\text{ is not a Fredholm operator}\}.
$$
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> $Ae_n=e_n/n$ 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.
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