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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 *point spectrum* of $T$ is the set\n$$\\sigma_p(T) = \\set{\\lambda \\in \\C : \\ker(T - \\lambda I) \\ne \\set{0}},$$\nthe set of [[eigenvalues|0BQAA2BK3Q]] of $T$.\n\n## Remarks\n\nHere $\\ker$ is the [[kernel|0S23E6YE83]].\nThe point spectrum is a subset of the [[spectrum|BMKATJ802A]] $\\sigma(T)$, and for $X = \\C^n$ the two are equal.\n\nExample: let $L(x_1, x_2, \\ldots) = (x_2, x_3, \\ldots)$ be the left shift on $\\ell^2$.\nFor $\\abs{\\lambda} \u003c 1$, the vector $x = (1, \\lambda, \\lambda^2, \\ldots)$ is in $\\ell^2$ and $Lx = \\lambda x$, so $\\lambda \\in \\sigma_p(L)$.\nConversely $Lx = \\lambda x$ forces $x_{n+1} = \\lambda x_n$, so $x = x_1 (1, \\lambda, \\lambda^2, \\ldots)$, which is in $\\ell^2$ with $x \\ne 0$ only if $\\abs{\\lambda} \u003c 1$; thus $\\sigma_p(L)$ is the open unit disc.\n\nNon-example: for the diagonal operator $D(x_n) = (x_n / n)$ on $\\ell^2$, $0 \\in \\sigma(D)$ but $0 \\notin \\sigma_p(D)$, since $Dx = 0$ forces $x = 0$.\n\nThe point spectrum need not be closed, and it can be empty, as for the right shift on $\\ell^2$.\n",
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