Is it permissible for a person to take something from the possessions of someone who intentionally damaged something of theirs and did not compensate them, without that person's knowledge and with the intention of compensation?

The matter presented in the question is one of two scenarios:

First: The reason for entitlement is apparent and does not require proof. In this case, it is permissible to take without permission. The evidence for this is that Hind bint Utbah took what sufficed her from her husband Abu Sufyan's wealth in a reasonable manner. Destruction (of property) is similar to usurpation (in this ruling).

Second: The reason for entitlement is not apparent, and there is no clear evidence. In this case, jurists have differed on two opinions:

The first opinion: It is not permissible for him to take. This is the doctrine of Malik and Ahmad, and it is the stronger opinion and the choice of Ibn Taymiyyah. Their evidence is the hadith: "Return the trust to the one who entrusted you, and do not betray the one who betrayed you." This is also because his honor might be impugned if someone else discovers the item he took.

The second opinion: It is permissible for him to take. This is the doctrine of Ash-Shafi'i. And according to Abu Hanifah, it is permissible to take from the same type as the right (owed).