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  "body": "\n## Statement\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $T \\colon H \\to H$ be a bounded linear operator with $T^* = T$, where $T^*$ is the [[Hilbert space adjoint|6HSJBJ8TTQ]].\nThen the [[spectrum|BMKATJ802A]] $\\sigma(T)$ is a subset of $\\R$.\nMore precisely, with $m = \\inf_{\\norm{x} = 1} \\ip{Tx}{x}$ and $M = \\sup_{\\norm{x} = 1} \\ip{Tx}{x}$, one has $\\sigma(T) \\subseteq [m, M]$, and both $m$ and $M$ belong to $\\sigma(T)$.\n\n## Remarks\n\nProof idea: let $\\lambda = a + ib$ with $a, b$ real and $b \\ne 0$.\nBecause $T - a I$ is self-adjoint, $\\norm{(T - \\lambda I)x}^2 = \\norm{(T - aI)x}^2 + b^2 \\norm{x}^2 \\ge b^2 \\norm{x}^2$.\nSo $T - \\lambda I$ is injective with closed range, and the orthogonal complement of its range is $\\ker(T - \\bar\\lambda I) = \\set{0}$ by the same estimate; hence $T - \\lambda I$ is invertible.\n\nExample: the multiplication operator $(Tf)(x) = x f(x)$ on $L^2[0,1]$ is self-adjoint, and its spectrum is $[0, 1]$, a real interval, although $T$ has no eigenvalues.\n\nSelf-adjointness is needed: the unilateral shift on $\\ell^2$ is not self-adjoint, and its spectrum is the closed unit disc.\nThe finite-dimensional case is the fact that [[eigenvalues of a self-adjoint operator are real|0C4NT4GT3X]].\nThe same conclusion holds for unbounded self-adjoint operators, and for self-adjoint elements of a C*-algebra.\n\n## Explanations\n\n### L1 human\n\nA self-adjoint operator is the infinite-dimensional analogue of a Hermitian matrix.\nIts spectrum, the set of numbers $\\lambda$ for which $T - \\lambda I$ cannot be inverted, lies on the real line.\nThis is why self-adjoint operators model measurable quantities in quantum mechanics.\n\n### L2 human\n\nIn infinite dimensions the spectrum can contain points that are not eigenvalues, so the finite-dimensional argument with an eigenvector does not suffice; the proof uses a lower bound instead.\nThe result is the first step towards the spectral theorem for self-adjoint operators, which represents $T$ as an integral of $\\lambda$ over $\\sigma(T) \\subseteq \\R$.\nThe numbers $m$ and $M$ are the ends of the closure of the [[numerical range|XWXZ0DGW3Z]], and $\\norm{T} = \\max(\\abs{m}, \\abs{M})$.\n",
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