Is there any hope of redemption ...?

[Aratlithiel]

In my view, all of Gollum's "evil" acts were perpetrated under the influence of the Ring. So why does he not deserve redemption?

While I understand the sympathy that many have for this character, the answer is actually very simple: Gollum never did anything to EARN his redemption. The definition of redemtion (redeem)in the Merriam-Webster Collegiate Dictionary is as follows;

REDEEM: to atone for : EXPIATE

The question now becomes, did Gollum atone for his "sins"? To atone means to take responsibility for and extinguish the guilt incurred by one's wrongs. Gollum did not do this. To blame his bad behavior and weak will on the Ring is to negate the very concept of atonement and redemption. He cannot redeem himself until he has taken responsibility for his actions and experienced remorse for them.

Accidentally saving the world while attempting to retrieve (by force) the one thing that can destroy it does not count toward redemption. Even in his last moments, Gollum does not feel sorrow or regret - he feels joy at having his Precious restored. The rest of the world means nothing to him - he cares only for himself and the Ring. In my opinion, this is not a good case for redemtion.