I have a large matrix $A$ of which I can't directly compute the transpose $A^T$. However, I have at my disposal a bad estimator of its transpose (let's call it $B$). I ultimately want to compute $A^Ty$, with $y$ and arbitrary vector. I was wondering if it is possible to, for example, define an iterated scheme $z_{k+1} = f(z_k),\ z_0 = By $ such that $\lim_{k \to \infty} z_k = A^Ty $, where $f$ would use multiplications by $A$ and $B$.
Thank you in advance.