Zero Operators on Complex Hilbert Space

Question

This is a problem from Kreyszig's Introdcutory Functional Analysis with Applications.

If for any $x$ in a complex Hilbert Space $<Tx, x> = 0$, show that $T\equiv 0$.

Any clue?

Answer 1

$\forall x, y$, we have following two equations:

\begin{align} <T(x+iy), x+iy> = 0 \\ <T(x+y), x+y> = 0 \end{align}

Since $<Tx, x> = <Ty, y> = 0$, these two are equivalent to \begin{align} <Ty, x> - <Tx, y> = 0 \\ <Ty, x> + <Tx, y> = 0 \end{align} Because $x$ and $y$ are chosen arbitrarily, we conclude that $T \equiv 0$