A factorization for operators

Question

Let $a$ be an arbitrary operator in $B(H)$ and $b$ be a positive operator in $B(H)$. Assume $a$ and $b$ have the same null space and there exists an operator $u\in B(H)$ with $a=ub$.

Q) Can we conclude the null space of $u$ is contained in the null space of $a$?

Answer 1

Take $b $ to be a nontrivial projection, $u $ a unitary, and $a=ub $. Then $\ker a\ne\ker u=\{0\} $.

For Q2, take $a=u=b $.