Is swearing an oath by triple divorce—for the purpose of reassuring the wife and putting an end to her doubts—that there is no other woman in the husband's life or that he will not approach what is forbidden, Islamically permissible, and what are its implications?

Swearing by divorce is impermissible. The fatwa we adhere to is that a conditional divorce with the pronouncement of "three" (triple divorce) takes effect as three divorces upon the condition's fulfillment, and the woman becomes irrevocably divorced (baynunah kubra).

If by your oath you intended to abstain from illicit relationships, then merely looking does not constitute a breach of the oath. However, if you intended to abstain from all forbidden acts with women, then a forbidden glance would breach the oath.

If the intention is absent, then the reference point is the reason for the oath and what prompted it, or the apparent meaning of the wording.

It appears that the oath should be interpreted based on its underlying cause, which is the existence of an illicit relationship with a woman. Therefore, merely looking does not breach the oath; rather, it is breached by establishing an illicit relationship with a woman.