Problem 6

Exercise 10.6 (Vanishing Condition on a Hilbert Space)   Let $H$ be a Hilbert space and let $A_n$ be a sequence of bounded linear operators on $H$. Assume for every $x,y\in H$ that $\lim \langle y, A_n x\rangle=0$.

1. Does it follow that $\lim \Vert A_n\Vert=0$?
2. Does it follow that $\sup \Vert A_n\Vert<\infty$?

Provide counterexamples or proofs.

Proof. For part (b), we provide a proof. Let $n$ be fixed. For any $x\in H$, set $y=A_nx$. The vanishing assumption indicates that $\langle A_nx, A_nx\rangle = 0$. $\qedsymbol$