The Shulhan Aruch in Siman 421 presents the Halachot of the Melacha of "Tohen"-grinding- on Shabbat. Maran rules that slicing vegetables thinly is prohibited as Tohen. From this it seems that it would be problematic to cut an Israeli salad on Shabbat, since the vegetables are cut in small pieces.
However, Hacham Ovadia explains this ruling based on Maran’s commentary in the Bet Yosef. There, Maran cites the opinion of the Rashba that Tohen is only prohibited when done to prepare for consumption at a later time. If done for immediate consumption, such as for the upcoming meal or to eat right away, Rashba rules that it is permitted. The Bet Yosef adds that no authority disagrees with this Rashba. Thus, it can be assumed that this Rashba is accepted as Halacha.
Based on this, Hacham Ovadia interprets Maran in Shulhan Aruch to mean that it is only prohibited to slice vegetables for a later time, but for immediate consumption the Shulhan Aruch would permit it, based on the Rashba. This interpretation is strengthened by the fact that the Rema explicitly adds to Maran’s ruling that when the chopping is for immediate consumption, it is permitted. The Rema does not present his ruling as a "Yesh Omrim"-opposing opinion, rather as an explanation of Maran’s words.
Therefore, Hacham Ovadia rules that it is permissible to cut an Israeli salad for immediate consumption or for the upcoming meal (i.e. within one half hour), on a regular cutting board with a regular knife. In Halichot Olam (Parashat Mishpatim) he adds that if one is stringent to cut the salad in bigger pieces, Tavo Alav Beracha-it is praiseworthy.
If someone mistakenly cut the vegetables several hours in advance of consumption, does the salad become forbidden to eat? Hacham Ovadia rules that while clearly the person was wrong to do so, yet it does not become forbidden, because the minority opinion of the Rosh may be relied upon. The Rosh holds that the prohibition of Tohen does not apply whatsoever to edible foods, only to grinding grain and so forth.
It is permitted to make an Israeli salad or smash a banana or avocado on Shabbat within half an hour of consumption. If one did not follow this Halacha and prepared the food before that, it does not become forbidden, B’diavad.