Let us assume that the current price $P$ is lower than the 'equilibrium price' $P^\star$ so that $Q$ is lower than $Q^\star$. If we move from this combination towards the equilibrium one, it may be the case that the producer surplus decreases. I understand that the total surplus (consumers+producers) increases, but it is not clear to me why producers should be willing to reduce their surplus. So it looks like to me that there is not pareto improvement in moving from $(P,Q)$ to ($P^\star$,$Q^\star$), in that consumers are better off but producers may be worse off. What is wrong with my reasoning? Could you help me with this, please?
A situation can be Pareto efficient without also being a Pareto improvement over every other situation.
So, here for example, equilibrium is Pareto efficient, but -- as you have noted -- is not a Pareto improvement over every other situation.
Example. Say I have 100 apples to divide between A and B. The allocation (60, 40) is Pareto efficient, but it is not a Pareto improvement over the allocation (59, 41).