The Gemara in Masechet Shabbat (150b) establishes the rule that one may not speak on Shabbat about a Melacha – an activity that is forbidden on Shabbat. Thus, for example, one may not say on Shabbat, “Tomorrow I will cook such-and-such food.” Since cooking is forbidden on Shabbat, it is forbidden to speak about cooking on Shabbat.
The exception to this rule, as the Gemara cites from the sage Shemuel, is speaking about an activity which, though generally forbidden on Shabbat, could be performed in a permissible manner on Shabbat. Shemuel gives the example of speaking of one’s plans to walk to a certain place situated at a distance from the city. Even though Halacha forbids walking a distance outside one’s city on Shabbat, one can avoid this prohibition by setting up small “Burganim” (“huts”) along the road. Shemuel thus ruled that since there is a permissible way to travel outside the city, it is permissible to speak of this travel on Shabbat.
The Rashba (Rabbi Shlomo Ben Aderet of Barcelona, 1235-1310) applies the Gemara’s ruling to a case of a person who accepted Shabbat early, as is commonly done during the summer months. According to the Rashba, a person who accepted Shabbat early may perform Melacha (before sundown) by undergoing the process of “Hatarat Nedarim” – annulling his vow. He can appear before a Rabbinical court and express his regret over having taken this “vow” to begin Shabbat early, and his acceptance of Shabbat is thereby voided. (This Halacha is codified by several later authorities, including the Bet Yosef and the Ben Ish Hai.) As such, the Rashba contends, speaking of Melacha during the period of “Tosefet Shabbat” – the time one voluntarily added onto Shabbat – is analogous to the Gemara’s case about speaking of traveling outside the city. Since there is a permissible way of performing the given act, it is permissible to speak about it on Shabbat. Hence, the Rashba concludes, if one accepted Shabbat early he may ask his fellow, who has not yet accepted Shabbat, to perform Melacha on his behalf. For example, if after the early Minyan on Friday evening a person realized that he had forgotten something in his car, he may ask somebody who has not yet accepted Shabbat to retrieve the item from the car and bring it to his home.
Summary: A person who accepted Shabbat early, before sundown, may ask somebody who has not yet accepted Shabbat to perform Melacha on his behalf.