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  "body": "\n## Summary\n\nA unitary operator is a bounded linear operator $U$ on a Hilbert space with $U^*U = UU^* = I$.\nEquivalently, it is a surjective linear map that preserves the inner product.\nUnitary operators are the isomorphisms of Hilbert spaces, so two operators that differ by conjugation with a unitary operator have the same properties.\nThey extend [[unitary matrices|0AR68JTMDT]] to infinite dimensions and describe the time evolution in quantum mechanics.\n\n## Explanations\n\n### L1 human\n\nA unitary operator is a rigid motion of a Hilbert space that can be undone.\nIt keeps all lengths and all angles, and every vector is hit by it.\nThe Fourier transform on $L^2(\\R)$, suitably normalized, is a unitary operator.\nA simpler example is the shift on two-sided sequences, $(Ux)_n = x_{n-1}$ on $\\ell^2(\\Z)$: it moves every entry one step to the right and is undone by the shift to the left.\n",
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