In a Hilbert Space, why is a bounded linear operator self adjoint if and only if it is symmetric?

Question

Equivalently, why does a bounded linear operator $A$ satisfy $A = A^{*}$ if and only if $\langle Ax, y\rangle = \langle x, Ay \rangle$? The first direction (assuming $A = A^{*}$) is obvious, but I do not see how to show the other direction.

Answer 1

The adjoint $A^*$ always satisfies $\left< Ax,y\right>=\left<x,A^*y\right>$. If then always $\left< Ax,y\right>=\left< x,Ay\right>$, then $\left< x,Ay\right>=\left< x,A^*y\right>$ or equivalently $\left< x,(A-A^*)y\right>=0$. Take $x=(A-A^*)y$ to get $(A-A^*)y=0$.