Let $T$ be a linear operator on an inner product space. I really just want a hint as to how prove that $N(T^{\dagger}\circ T) = N(T)$, where "$^\dagger$" stands for the conjugate transpose.
Just as an aside, how should I read to myself the following symbolism:
Hint: Let $V$ denote your inner product space. Clearly $N(T)\subseteq N(T^* T)$, so you really want to show that $N(T^* T)\subseteq N(T)$. Suppose $x\in N(T^* T)$. Then $T^* Tx = 0$, so we have $\langle T^* Tx, y\rangle = 0$ for all $y\in V$. Can you see where to go from here?