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  "body": "\n## Definition\n\nLet $X$ be a complex Banach space with identity map $I$, and let $T \\colon X \\to X$ be a bounded operator.\nThe *essential spectrum* of $T$ is the set\n$$\\sigma_{ess}(T) = \\set{\\lambda \\in \\C : T - \\lambda I \\text{ is not a Fredholm operator}}.$$\nHere a bounded operator is [[Fredholm|8HWGKWD6ZM]] when its kernel is finite-dimensional and its image is closed and of finite codimension.\n\n## Remarks\n\nSince every invertible operator is Fredholm, $\\sigma_{ess}(T)$ is a subset of the [[spectrum|BMKATJ802A]] $\\sigma(T)$; it is compact.\nIt does not change when a [[compact operator|M2NEP1K5AE]] is added to $T$.\nOn a Hilbert space, it is the spectrum of the image of $T$ in the quotient algebra of bounded operators by compact operators, by Atkinson's theorem.\n\nExample: if $X$ is infinite-dimensional, then $1 \\in \\sigma_{ess}(I)$, since $I - I = 0$ has an infinite-dimensional kernel.\n\nNon-example: for $D(x_n) = (x_n / n)$ on $\\ell^2$, $1 \\in \\sigma(D)$ but $1 \\notin \\sigma_{ess}(D)$.\nThe kernel of $D - I$ is spanned by $e_1$, and its image is the closed subspace $\\set{x : x_1 = 0}$ of codimension $1$, so $D - I$ is Fredholm.\nIn fact $\\sigma_{ess}(D) = \\set{0}$.\n\nIn Banach spaces several non-equivalent variants of the essential spectrum are in use.\nFor a self-adjoint operator on a Hilbert space all the usual variants agree.\nThen $\\lambda \\in \\sigma_{ess}(T)$ if and only if $\\lambda$ is in $\\sigma(T)$ and is not an isolated eigenvalue of finite multiplicity.\n",
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