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  "body": "\n## Definition\n\nLet $X$ be a complex Banach space, and let $T \\colon X \\to X$ be a bounded operator with [[spectrum|BMKATJ802A]] $\\sigma(T)$.\nA point $\\lambda \\in \\sigma(T)$ is a *normal eigenvalue* of $T$ if:\n\n1. $\\lambda$ is an isolated point of $\\sigma(T)$, and\n2. the [[Riesz projector|HNW4WRSEEX]] $P_{\\set{\\lambda}}$ of $T$ for $\\set{\\lambda}$ has finite rank.\n\nThe *discrete spectrum* $\\sigma_{disc}(T)$ is the set of normal eigenvalues of $T$.\n\n## Remarks\n\nA normal eigenvalue is an eigenvalue, and its algebraic multiplicity, the rank of $P_{\\set{\\lambda}}$, is finite.\n\nExample: if $T$ is a [[compact operator|M2NEP1K5AE]], then every nonzero point of $\\sigma(T)$ is a normal eigenvalue.\nFor $D(x_n) = (x_n / n)$ on $\\ell^2$, each $1/n$ is isolated in $\\sigma(D)$, and $P_{\\set{1/n}}$ is the projection onto the span of $e_n$, of rank $1$.\n\nNon-example: if $X$ is infinite-dimensional, then $\\sigma(I) = \\set{1}$ and $1$ is isolated, but $P_{\\set{1}} = I$ has infinite rank, so clause 2 fails.\nFor $D$ above, $0$ fails clause 1, since $1/n \\to 0$.\n\nFor a self-adjoint operator $T$ on a Hilbert space, $\\sigma_{disc}(T)$ is the set of isolated eigenvalues of finite multiplicity.\nThen $\\sigma(T)$ is the disjoint union of $\\sigma_{disc}(T)$ and the [[essential spectrum|FRZG0E15SK]].\n",
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