[math] \newcommand{\mathds}{\mathbb}[/math]

Prove that an operator [math]T\in B(H)[/math] satisfies the condition

[[math]] \lt Tx,x \gt \geq0 [[/math]]

for any [math]x\in H[/math] precisely when it is positive in our sense, [math]\sigma(T)\subset[0,\infty)[/math].