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  "body": "\n## Summary\n\nThe adjoint of a densely defined operator $T$ between Hilbert spaces is an operator $T^*$ in the opposite direction.\nIts domain consists of the vectors $y$ for which $x \\mapsto \\ip{Tx}{y}$ is bounded on the domain of $T$.\nIt extends the [[Hilbert space adjoint|6HSJBJ8TTQ]] of bounded operators, and the domain of $T^*$ is part of the answer.\nThe adjoint is needed to define self-adjoint unbounded operators, the observables of quantum mechanics.\n\n## Explanations\n\n### L1 human\n\nFor bounded operators, the adjoint moves the operator from one side of the inner product to the other.\nFor an unbounded operator this can only be done for some vectors $y$, and the adjoint is defined exactly on those.\nThe domain of $T$ must be dense; otherwise the vector $T^* y$ would not be determined.\nFor the diagonal operator $(x_n) \\mapsto (n x_n)$ on its natural domain in $\\ell^2$, the adjoint is the same operator on the same domain.\n",
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