Gemara

(Beraisa): If one wrote a big letter in the space needed to write two [normal] letters, he is exempt;

If one erased a letter big enough to write two letters in its place, he is liable;

R. Menachem b'Rebbi Yosi says, this is a stringency of erasing over writing.

92a (Mishnah): One is liable for Hotza'ah whether he was Motzi in his right or left hand, in his garment or on his shoulder, because this is how Benei Kehas carried.

103a (Mishnah): If one writes two letters he is liable, whether he used his right or left hand.

Question: Why is he liable for writing with the left hand? This is abnormal!

Answer #1 (Abaye): The Mishnah discusses an ambidextrous person.

Answer #2 (Rav Yakov): Our Mishnah is like R. Yosi, who is Mechayev for two letters due to Roshem (marking).

Tosefta (Shabbos 10:11): One who detaches with his right or left hand is liable.

Rishonim

Rambam (Hilchos Shabbos 11:14): One who writes with his left hand is exempt.

Rambam (17): Roshem is a Toladah of writing. One who draws impressions and forms on a wall or lacquer, like artists do, is liable for writing. Likewise, one who erases an impression in order to fix is liable for a Toladah of erasing.

Magid Mishneh: Perush ha'Mishnayos says that R. Yosi holds that Roshem is another Av. Writing just two letters is liable for Roshem. The Halachah follows Chachamim, that it is a Toladah of writing. Rashi explained differently.

Ohr Same'ach: It is not clear whether or not Roshem with the left hand is liable. The Rambam connotes that it is a Toladah of writing, so it is like writing. Regarding the Shi'ur, the Toladah is unlike the Av. One is liable for melting any amount of metal, even though it is a Toladah of cooking, for which the Shi'ur is k'Grogeres (the size of a fig). This is due to metal's importance. One is liable for any amount of metal also for Hotza'ah. However, we do not find that Avos and Toladah differ about liability for the left hand. R. Yosi holds that Roshem is an Av, therefore one is liable for the left hand. However, the Rambam could hold that the first Tana agrees, for the Toladah is not exactly like the Av. It is normal to mark with the left hand. However, writing two letters is the Av. Just like the Shi'ur of cooking is k'Grogeres, also writing two letters. It has all laws of the Av. Since it is not normal to write with the left hand, he is exempt. It seems that this is correct. This requires investigation.

Rambam (12:12): One is liable for Hotza'ah whether he was Motzi in his right or left hand... for this is normal. One is liable for Hotza'ah on his shoulder, because this is how Benei Kehas carried for the Mishkan, above 10 Tefachim.

Rashi (92a DH she'Chen): From Kehas we learn carrying on the shoulder. We know that one is liable for carrying in his right or left hand or in his garment, for these are normal. Rav Hai Gaon brings from the Yerushalmi that Elazar used to carry four things at once in these four ways.

Chidushei (what was initially attributed to) ha'Ran (Shabbos 103a DH Amar): Why is one exempt for writing with the left hand? One is liable for Hotza'ah with the left hand! I answer that Hotza'ah is common with the left hand just like with the right hand, but writing is a delicate Melachah. It is abnormal to write with the left hand.

Poskim and Acharonim

Chasdei David (on Tosefta 10:11): The Tosefta teaches that the left hand is not a Shinuy [for detaching]. Only with the back of his hand or foot is a Shinuy.

Chazon Yechezkeil: This is only for detaching, which is like Hotza'ah. Any Melachah normally done with one hand, surely it is normal to do it only with the right hand. If he used the left, this is abnormal, and he is exempt.

Kaf ha'Chayim (340:44, citing Eliyahu Rabah): The Rambam (11:14) exempts one who wrote with his left hand. It seems that the same applies to erasing. Therefore, it is good to use the left hand to break cakes [with writing on them] or open a Sefer [with writing on the edge of the pages, which is broken when one opens the Sefer].

Rebuttal (Avnei Nezer OC 209:19): One is liable for erasing with the left hand, just like for Hotza'ah, or for writing according to R. Yosi, who obligates for Roshem with the left hand. We exempt only for writing, for one cannot write well with the left hand.

Yabi'a Omer: Eliyahu Rabah overlooked the Ran, who says that this applies [only] to writing, which is a fine Melachah. If not, he is liable with the left hand. Zekan Aharon (1:15) says that one cannot write with his left hand as nicely as with his right, so he is exempt with the left. This does not apply to milking and other Melachos. He challenged Eliyahu Rabah, but suggested that erasing requires precision, not to erase the adjacent letters. One cannot do so with his left hand. However, breaking a cake is the same with either hand. One is no more exempt with the left than with the right. Perhaps Eliyahu Rabah holds that since one is liable for erasing only if he intends to write, it is judged like writing. Chemdas Yisrael was unsure whether one is liable for erasing with the left hand. Maharsham (Orchos Chayim 340:5) distinguishes right from left for writing and erasing, but not for other Melachos. Itim l'Binah learns from the Tosefta to obligate for all other Melachos with the left hand. There is no reason to exempt one who puts a pot on the fire with the left hand. Perhaps even Eliyahu Rabah said so only for a cake and a Sefer, for there are other reasons to be lenient.

Minchas Chinuch (32, Musach Shabbos Kosev (34), 8): One is exempt for writing with the left hand, for it is not Derech Kesivah. It is like writing k'Le'acher Yad (unskillfully), which is exempt on Shabbos. Even so, perhaps it is Kosher for a Sefer Torah, Mezuzah, Tefilin or Get. K'Le'acher Yad applies only to Shabbos. The Rema (EH 123:1) is Machshir a Get b'Di'eved. The Beis Yosef brings Sefer ha'Terumah, who say that it is unreasonable that k'Le'acher Yad is not writing for Shabbos, but in general it is writing. If one cast down a knife [and it cut the Simanim] it is a Kosher Shechitah (Chulin 31a), even though he is exempt for Shabbos, since it is k'Le'acher Yad. Perhaps one is exempt for writing Stam matters on Shabbos with the left hand, but if one wrote a Sefer Torah, Mezuzah or Tefilin, since it is writing for this matter, it is writing regarding Shabbos.

Minchas Chinuch (ibid., 15): Roshem has the law of writing in every way, since it is a Toladah of it. Only R. Yosi obligates for the left hand. I do not understand Perush ha'Mishnayos.

Chayei Adam (2:9:2, cited in Mishnah Berurah 340:22): One is liable for Melachah only in the normal way to do it. It seems that whether one used his right or left hand he is liable, except for writing, which is exempt with the left hand. (The Tosefta proves this.)

Tif'eres Yisrael (before Maseches Shabbos, Kupas ha'Rochlim 3): One who writes with the left hand is exempt, for this is not Derech Kesivah. This is only for writing. It is normal to write only with the right hand, so this is a Shinuy. Other Melachos are different. E.g. one who lights a fire with either hand is liable, for sometimes one lights with the right hand, and sometimes with the left. The Gemara says so about Hotza'ah (92a).

Tif'eres Yakov (on Tif'eres Yisrael): Rav Hai Gaon connotes that we would not know to obligate Hotza'ah with the left hand had Elazar not carried this way. However, Rashi's first Perush and the Rambam are like the Tif'eres Yisrael. The Mishnah and Rambam mentioned Chiyuv with the left hand only regarding Hotza'ah. Perhaps it is not so (it seems that this should say 'exempt' - PF) for other Melachos, except for writing. Tosfos (Shabbos 2a DH Yetzi'os) said that Hotza'ah is a weak Melachah. One might have thought that together with another weakness, e.g. using the left hand, one is exempt. The Mishnah teaches that this is not so, and all the more so one is liable for other Melachos with the left hand.

Bi'ur Halachah (340:5 DH Mutar): The Magid Mishneh is astounding. Writing two letters is proper writing. It is not a mere Toladah! The Rambam does not discuss mere scratches. He rules unlike R. Yosi regarding letters! We have no proof that Rabanan hold that there was Roshem in the Mishkan! The Rambam explains that he discusses Roshem of designs, [like the Yerushalmi says]. He exempts one who writes with the left hand. He, and many Tana'im, obligate for two letters only if they form a word (Halachah 10). They hold that one is exempt for mere scratches!

Divrei Yechezkeil (4:2): The Rambam holds that the first Tana agrees that one is liable for writing with the left hand. If he is ambidextrous, he is liable for writing. If not, he is liable for Roshem, which is a Toladah of writing. Why does the Minchas Chinuch says that the first Tana exempts? In 11:14, the Rambam obligates for writing only with the right hand. This implies that he is exempt with the left, even for Roshem. This is because he intended to write, and not to mark. In Halachah 17, regarding Roshem, the Rambam did not specify the right hand. It is like all other Melachos.

Afikei Yam (4:5): There is no reason why a Toladah cannot be more stringent than the Av. K'Le'acher Yad exempts every Melachah and its Toldos. R. Yosi obligates Roshem with the left hand. This shows that even though it is abnormal for writing, it is normal for Roshem. Rashi (Sukah 37b DH Aval) obligates for Tolesh (detaching) with his mouth, even though this is a Toladah of Kotzer (harvesting). Surely one is exempt for harvesting with his mouth! A Beraisa gives a stringency of erasing over writing. Why didn't it add that one is liable for erasing with the left hand, but exempt for writing? Rather, this is not a stringency. (Everything is liable in the normal way.)

R. E.M. Horovitz (Pesachim 66a): Why were Chachamim unsure whether we offer Korban Pesach on Shabbos (Pesachim 66a)? The Pesach offered in the Midbar [after Chanukas ha'Mishkan] was on Shabbos [for Rosh Chodesh Nisan was on Sunday - Shabbos 87b]! One can offer Pesach without Chilul Shabbos mid'Oraisa, e.g. to slaughter with the left hand. They were unsure whether we [who forbid Melachah through a Shinuy] permit.