The Shulhan Aruch (Orah Haim 263) rules (based on the Rashba) that if a person accepted Shabbat early, before sunset – as one is supposed to do – he is allowed to ask another Jew, who has not accepted Shabbat, to perform Melacha on his behalf. As long as the sun has not yet set, one who has accepted Shabbat may have somebody else perform Melacha for him, even though he himself obviously may not perform Melacha, as he has accepted Shabbat. Since the other person has not yet accepted Shabbat, and the person who had accepted Shabbat would be allowed to perform Melacha were it not for his decision to accept Shabbat early, he may ask somebody else to perform Melacha which he needs to be done.
One example of this Halacha is a case where a woman needs to go to the Mikveh on Friday night. She may light the Shabbat candles and accept Shabbat, and then her husband can drive her to the Mikveh. As long as she does not open doors or perform any other action that activates lights or electronic devices, she may have her husband drive her to the Mikveh. Conversely, if the husband accepted Shabbat and the wife has not yet accepted Shabbat, she may drive him to the synagogue. Likewise, if the husband accepted Shabbat and prayed in the synagogue before sundown, as many men do in the summertime, and he comes home before his wife lit candles and accepted Shabbat, he may ask his wife to turn on a light, turn on the air conditioner, or perform other activities involving Melacha.
This is the ruling of the Shulhan Aruch, as discussed by Rabbi Moshe Halevi (Israel, 1961-2001) in his Menuhat Ahaba (vol. 1, p. 111; listen to audio recording for precise citation).
Summary: One who accepts Shabbat before sundown may ask another Jew who had not yet accepted Shabbat to perform Melacha on his behalf, such as to turn on a light or air conditioner, as long as the sun has not yet set.
