Prove that operator is Zero operator.

Question

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..!!

Answer 1

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 $y$.