My question is targeted towards a very particular detail in my research that I am trying to understand. I will therefore break it down into some more general questions.

Let $X$ be a smooth variety, $i:D\hookrightarrow X$ a smooth divisor. Let $E$ be a holomorphic vector bundle, $E|_D =i_*i^* E$ and $\mathcal{A}_X$ the sheaf of smooth functions on $X$.

The tensor product $E^{\infty}=E\otimes_{\mathcal{O}_X}\mathcal{A}_X$ is the sheaf of smooth sections of $E$ as a vector bundle. How do I understand the sheaf $E|_D\otimes_{\mathcal{O}_X}\mathcal{A}_X$?

Let $\rho: E\to E|_D$ be the restriction of holomorphic sections. Consider $\rho\otimes\text{id}:E^{\infty}\to E|_D\otimes_{\mathcal{O}_X}\mathcal{A}_X$. Is there some way to understand how this map acts on the smooth sections of $E$?

Now the actual question that I would like to understand.

- Let $T$ be the tubular neighborhood of $D$, $\phi:E|_D\to E|_D$ an automorphism of the vector bundle on $D$. Let $\tilde{\phi}: E^\infty|_T\xrightarrow{\sim} E^\infty|_T$ be a smooth extension of $\phi$ to $T$ (as $T$ is smoothly retractible to $D$). Does the following diagram commute?

Clearly, if $T$ is a holomorphic tubular neighborhood, then question 3 would have a positive answer. But existence of a such neighborhood is very restrictive. Therefore, I am hoping that understanding the first two questions, will help me better understand the last one. I appreciate any comments, corrections and suggestions.