Suppose $n \geq p \geq m$. Let $A \in \mathbb{R}^{m \times n}$ be of full row rank and $B \in \mathbb{R}^{n \times p}$ be of full column rank. If $A B$ is an $m$-by-$p$ matrix of full row rank. Can we have an estimation of $\|(A B)^{\dagger}\|$ with respect to $\|A^{\dagger}\|$ and $\|B^{\dagger}\|$ (and maybe $\|A\|$ and $\|B\|$)?
We can write \begin{equation*} (A B)^{\dagger} = B^{*} A^{*} (A B B^{*} A^{*})^{-1}. \end{equation*} But I have no idea how to bound the inverse term.
Any advice is appreciated. Very thanks!