TITLE: Prove that operator is Zero operator.
QUESTION [6 upvotes]: Let $H$ be the Hilbert space and $T$ be bounded linear operator on $H$. If $$\langle T^2x,x\rangle =0, \forall x \in H \quad \text{and} \quad \langle Tx,x\rangle =0, \forall x \in H, $$
then $T=0$.
I thought so much about this problem but could not get any clue to tackle this problem. Someone give the hint to solve this one thank you..!!

REPLY [5 votes]: Hint : It is enough to prove that
$$\langle Tx,Tx\rangle=0.$$
To show this, you can add suitable terms that are orthogonal to $Tx$ into the inner product to obtain a term of the form $\langle Ty,y\rangle$ for some $y$.