Positive operators in Hilbert spaces

Question

Let $H$ be a Hilbert space. I am just asking if there's some reference which studies operators $A$ with this property:

$$\left\langle Ax,x\right\rangle \geq0,$$ for all $x\in H$. And $Ax=0$ whenever

$$\left\langle Ax,x\right\rangle =0.$$

And maybe some applications of these operators on differential equations in Hilbert spaces.

Answer 1

If you are dealing with complex Hilbert spaces, then the first condition implies the second one. Such operators are called positive operators and they are considered in most functional analysis texts and all operator algebras texts.

Answer 2

If $A$ is positive, then the condition $\langle Ax,x\rangle=0$ implies $Ax=0$.

Indeed, every positive operator $B$ on a Hilbert space admits a unique positive root $\sqrt B$ that satisfies $(\sqrt B)^2=B$. In your case this yields $$||\sqrt Ax\|^2=\langle \sqrt{A}x,\sqrt{A}x\rangle=\langle Ax,x\rangle=0.$$ Therefore we have $\sqrt Ax=0$ and finally $Ax=\sqrt A(\sqrt Ax)=0.$