If $\Bbb H$ is a $\Bbb C$-Hilbert space and $A\in \Bbb B(\Bbb H),$i.e. bounded linear operator on $\Bbb H$ such that $\langle Ah,h\rangle=0$ for all h in $\Bbb H$, then $A=0$
For the proof of this proposition, why we don't need the extra assumption $A=A^*?$ When I have this extra assumption I know $\|A\|=\sup\{|\langle Ah,h \rangle: \|h\|=1\}$, then done very easily. Could you please help me?
Thanks in Advance.
First note that if $\langle Ah, h'\rangle=0$ for all $h,h'\in\mathbb H$ then $A=0$. Recall the polarization identity $$\langle h,h'\rangle=\frac{1}{4}(\langle h+h',h+h'\rangle - \langle h-h',h-h'\rangle +\langle h+ih',h+ih'\rangle-\langle h-ih',h-ih'\rangle$$ It follows from the first observation and the polarization identity that $\langle Ah,h\rangle=0$ for all $h$ implies $A=0$.