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  "body": "\n## Summary\n\nA positive operator is a bounded self-adjoint operator $A$ on a Hilbert space with $\\ip{Ax}{x} \\ge 0$ for every vector $x$.\nIt is the operator version of a [[positive semidefinite matrix|01MMVW8CKC]] and of a nonnegative real number.\nPositive operators define an order $A \\le B$ on self-adjoint operators, they have [[square roots|3JY5HNFRMF]], and every operator of the form $T^*T$ is positive.\n\n## Explanations\n\n### L1 human\n\nA positive operator never turns a vector to point against itself: the component of $Ax$ along $x$ is never negative.\nOn $\\C^2$ the diagonal matrix with entries $1$ and $2$ is positive, since $\\ip{Ax}{x} = \\abs{x_1}^2 + 2\\abs{x_2}^2 \\ge 0$.\nThe diagonal matrix with entries $1$ and $-1$ is not, since it sends $(0, 1)$ to $(0, -1)$.\nPositive operators play the role of the nonnegative numbers among operators.\n",
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