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  "body": "\n## Definition\n\nLet $H$ be a [[Hilbert space|HM2HHE3J51]] and let $A \\colon H \\to H$ be a [[positive operator|77020CY5ZW]].\nA *positive square root* of $A$ is a positive operator $B \\colon H \\to H$ with $B^2 = A$.\nEvery positive $A$ has exactly one positive square root; it is called *the square root* of $A$ and written $A^{1/2}$ or $\\sqrt{A}$.\n\n## Remarks\n\nExample: on $L^2[0, 1]$ let $A$ be multiplication by $x$, so $(Af)(x) = x f(x)$.\nMultiplication by $\\sqrt{x}$ is a positive operator $B$, since $\\ip{Bf}{f} = \\int_0^1 \\sqrt{x} \\abs{f(x)}^2 \\, dx \\ge 0$, and $B^2 f = x f$; so $A^{1/2} = B$.\n\nNon-example: $-I$ satisfies $(-I)^2 = I$, but $\\ip{-x}{x} = -\\norm{x}^2 \u003c 0$ for $x \\ne 0$, so $-I$ is a square root of $I$ but not the positive square root.\n\nThe square root is given by the continuous functional calculus, $A^{1/2} = f(A)$ with $f(t) = \\sqrt{t}$ on the spectrum of $A$.\nIt is also the norm limit of polynomials in $A$, so it commutes with every bounded operator that commutes with $A$.\n",
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