Hmm, interesting, this topic appeared on another translation exercise (το μισό πρωινό if I remember correctly), and I was confused, αλλά τώρα καταλαβαίνω, ευχαριστώ! (though I must say, if I were in Greece and had to ask for "half of the sandwich" I would probably say "το μισό του σάντουιτς" (even if it is not common) just to ease my linguistic conscience...)
No, at least not in English (or at least not in my dialect of English- I'm from the United States). To me, "the half sandwich" means a portion/serving of sandwich that is half the normal sandwich size whereas "half of the sandwich" is the half part of a whole sandwich. I suppose I am influenced by restaurant experiences here in the US because on many menus there is the option to order a "half sandwich" and that is how it would be ordered (= "I would like the half sandwich, please.") If one said "I would like half of the sandwich" that would imply that someone else already has the whole sandwich and you are asking them for half of it. I admit it is a very subtle difference, but one that exists nevertheless (at least in my brain). I am not saying it has to be that way in Greek, but just thought I would explain the difference in English. By the way, thank you so much for your explanations- they are really helpful.
Oh, I see. So, according to the meanings you gave me The half sandwich is Μισό σάντουιτς (I want the half sandwich=Θέλω μισό σάντουιτς) and half of the sandwich is το μισό σάντουιτς (I want half of the sandwich=Θέλω το μισό σάντουιτς). I didn't knew that there was such a difference in english between the two, and seems like word-to-word translation just complicates things in this case. Το μισό του σάντουιτς is not really used but will get your point across and would mean "half of the sandwich" but it's the definite articles that make it have that meaning and not the genitive case.