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  "body": "\n## Definition\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $U \\colon H \\to H$ be a bounded linear operator with [[adjoint|6HSJBJ8TTQ]] $U^*$.\nThe operator $U$ is *unitary* when\n$$U^* U = U U^* = I,$$\nwhere $I$ is the identity operator of $H$.\n\n## Remarks\n\nExample: on $\\ell^2(\\Z)$ the bilateral shift $(Ux)_n = x_{n-1}$ has adjoint $(U^*x)_n = x_{n+1}$, and both $U^*U$ and $UU^*$ are the identity, so $U$ is unitary.\n\nNon-example: on $\\ell^2(\\N)$ the unilateral shift $S(x_0, x_1, \\ldots) = (0, x_0, x_1, \\ldots)$ satisfies $S^*S = I$, but $SS^*$ sends $(1, 0, 0, \\ldots)$ to $0$, so $SS^* \\ne I$ and $S$ is not unitary.\nIt is an isometry that is not surjective.\n\nEquivalent forms: $U$ is unitary if and only if $U$ is surjective and $\\ip{Ux}{Uy} = \\ip{x}{y}$ for all $x, y$, and if and only if $U$ is a surjective isometry.\nIn finite dimensions $U^*U = I$ already implies $UU^* = I$, but in infinite dimensions both conditions are needed, as the unilateral shift shows.\nThe spectrum of a unitary operator lies on the unit circle.\n",
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