Is it permissible to help a friend's mother manage her Islamic library and deal with men on her behalf, knowing that her husband has passed away?

The private seclusion of a man with a non-mahram (definition) woman is forbidden by Islamic law, based on the Prophet's (PBUH) sayings: "No man should be in seclusion with a woman except if she has a mahram with her," and "Beware of entering upon women," and "No man is in seclusion with a woman but the third of them is Satan." Therefore, if your assistance to this lady leads to seclusion, it is forbidden; otherwise, there is no harm. For the absence of seclusion, it is stipulated that there be a child present who possesses awareness and discernment. It is permissible for a woman to interact with men when there is a need, provided she adheres to the Islamic guidelines.