Positive operator ($T\ge 0$)

A bounded Hermitian operator $T$ on a Hilbert space is positive when $⟨Tx,x⟩\ge 0$ for every $x$. Its spectrum is contained in $[0,\infty )$.

Every bounded positive operator $T$ has a unique positive operator $T^{1/2}$ satisfying $(T^{1/2}{)}^2=T$. The spectral theorem for normal operators on a separable Hilbert space constructs it by applying the scalar square-root function to the spectrum.