prove or disprove if $\|AB - I\|<1$ then $\|BA - I\|<1$ can we say if $\|AB - I\|<1$ then $\|BA - I\|<1$ for some arbitrary norm.
I am trying to make counter example but I stuck please help me.

As the other answers nicely show we cannot say that your claim holds for all norms. But if we set $\| A \| = \max\{|\lambda_i|\ \ | \  \lambda_i \in \sigma(A) \}$, $\sigma(A)$ stands for the spectrum of $A$ , we can show that $\|AB \| = \|BA\|$. This is simply because $\sigma(AB) = \sigma(BA)$. Subtracting identity is just shift of the spectrum so $\sigma(AB-I) = \sigma(BA-I)$ and also $\| AB - I \| = \| BA - I \|$.
