What is the ruling regarding a man telling his wife: "By that which is unlawful from you, you shall not go to my family's house except one day a week," given that he does not recall his intention at the time of utterance, and is an expiation for an oath (kaffarat yameen) required of him?

Swearing by rendering one's wife unlawful (al-haram) can be interpreted as divorce (talaq), dhihar, or an oath (yamin). This is a matter over which scholars have differed. The most preponderant opinion is that this is an oath, and it does not necessitate divorce, even if one intended to swear by divorce. Since you have offered expiation for an oath (kaffarat yamin), we hope that suffices for you. However, you must be extremely cautious about toying with words of divorce and unlawfulness, and anything that could lead to the breakdown of the marital bond and falling into distress and regret.