Positive operator on a complex Hilbert space is symmetric?

Question

I've read somewhere that a positive operator on a complex Hilbert space is necessarily symmetric. I don't know how to prove this claim.

What I call a positive operator on $\mathfrak{h}$ is an operator $A : \mathcal{D}(A) \to \mathfrak{h}$ such that $\langle Ax, x \rangle \geq 0$ for every $x \in \mathcal{D}(A)$. Remember that here I'm only talking about symmetric-ness, not self-adjoint-ness !

Any help greatly appreciated !

Answer 1

We have $$ \mathbb{R}\ni\langle A(x+y),x+y\rangle=\langle Ax,x\rangle+\langle Ay,y\rangle+\langle Ax,y\rangle+\langle Ay,x\rangle. $$ Thus $\langle Ax,y\rangle+\langle Ay,x\rangle=\langle x,Ay\rangle+\langle y,Ax\rangle$.

Moreover, $$ \mathbb{R}\ni \langle A(x-iy),x-iy\rangle=\langle Ax,x\rangle+\langle Ay,y\rangle+i\langle Ax,y\rangle-i\langle Ay,x\rangle, $$ which implies $\langle Ax,y\rangle-\langle Ay,x\rangle=\langle x,Ay\rangle-\langle y,Ax\rangle$.

If you add up these to equalities, you arrive at $\langle Ax,y\rangle=\langle x,Ay\rangle$.