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  "body": "\n## Definition\n\n*Spectral theory* is the part of [[operator theory|YG9NR68ZH3]] that studies a linear operator $T$ on a Banach or Hilbert space through its [[spectrum|BMKATJ802A]] $\\sigma(T)$.\nThis is the set of complex numbers $\\lambda$ for which $T - \\lambda I$ has no bounded inverse.\nIt extends the theory of [[eigenvalues|0BQAA2BK3Q]] and [[eigenvectors|0N6960W0BV]] of matrices.\nIts main topics are the location and the parts of the spectrum, the resolvent and functional calculus.\nIt also contains the spectral theorems, which represent operators as multiplication operators.\n\n## Remarks\n\nExample: for the diagonal operator $D(x_n) = (x_n / n)$ on $\\ell^2$, spectral theory gives $\\sigma(D) = \\set{1/n : n \\ge 1} \\cup \\set{0}$.\nEach $1/n$ is an eigenvalue with eigenvector the $n$-th unit vector, and $0$ is in the spectrum without being an eigenvalue.\n\nNon-example: the study of the solutions of a nonlinear equation such as $u'' = u^3$ is not spectral theory in this sense, since no linear operator and no spectrum are involved.\nOnly its linearisation can be studied spectrally.\n\nThe word \"spectrum\" was used by David Hilbert before quantum mechanics.\nOnly later did it turn out that the spectral lines of atoms correspond to differences of eigenvalues of a quantum Hamiltonian.\n\n## Explanations\n\n### L1 human\n\nSpectral theory is the study of the \"eigenvalues\" of operators in infinite dimensions.\nIt explains what replaces eigenvalues and eigenvectors when an operator has too few of them.\n",
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