Let $X=(L^1([a,b]),\| \cdot \|_{L_1})$. Further let $ A \in B(X)$ a positive bounded operator. I know that X is complete. I want to to scale the norm of A by changing the norm of X, for example $$||f|| := ||fw||_{L_1} $$ for a positive weight function $w$. After scaling it should hold that $||A||<1$ and I want $ (L^1([a,b]),|| \cdot || )$ to be complete too.
I tried this with specific functions $w$ of which none worked out. Then I tried to do it as general as possible but I always end up with contradiction.
I want to know if this is even possible.