Positive operator

Remarks

Example: for every bounded operator $T$ on $H$, the operator $A = T^*T$ is positive. It is self-adjoint since $(T^*T)^* = T^*T$, and $\ip{T^*Tx}{x} = \ip{Tx}{Tx} = \norm{Tx}^2 \ge 0$.

Non-example: the diagonal operator $A = \mathrm{diag}(1, -1)$ on $\C^2$ is self-adjoint, but $\ip{A e_2}{e_2} = -1 < 0$ for $e_2 = (0, 1)$, so condition 2 fails.

Over $\C$, condition 2 alone implies condition 1, because a bounded operator with $\ip{Ax}{x}$ real for all $x$ is self-adjoint by polarization. Over $\R$, condition 1 is needed: a rotation by $90$ degrees of $\R^2$ has $\ip{Ax}{x} = 0$ for all $x$ but is not self-adjoint. Equivalent forms: $A$ is positive if and only if $A$ is self-adjoint with spectrum in $[0, \infty)$, and if and only if $A = B^*B$ for some bounded operator $B$. Some authors call these operators positive semidefinite or nonnegative.