The Shulhan Aruch in Siman 315 discusses the prohibition of making an Ohel (tent) on Shabbat. An Ohel is defined as space enclosed by partitions covered by a roof or overhanging to protect that which is underneath it. This protection includes, for example, providing shade from the sun or shelter from the rain. One of the most famous modern applications of this prohibition is opening and using an umbrella on Shabbat.
The Shulhan Aruch rules that a folding chair, known as a "Kiseh Traskal," may be opened on Shabbat. The Hazon Ish (Rav Abraham Yeshaya Karelitz, 1878-1953) derived from this a general principle that anything that is already built, but is merely folded or collapsed, may be opened on Shabbat, even though it forms an Ohel beneath it. Accordingly, it should follow that the Hazon Ish would permit opening an umbrella, since it is already fully assembled, and there are no pieces being added. Nevertheless, he rules that an umbrella may not be opened since it constitutes "Uvdin D’hol" (mundane activity). The Hatan Sofer also found reasons to be lenient, although his conclusion is to be strict.
The Bet Meir (Rabbi Meir Posner, 1729-1807) went so far as to actually permit opening an umbrella.
The Hida (Rav Haim Yosef David Azulai, 1724-1807) in his Birkeh Yosef (end of 315 in the Shiyureh Beracha) cites a Posek called the Giva’at Pinhas who prohibits opening the umbrella but allows carrying an umbrella that was already open before Yom Tob.
Hacham Ovadia in Yehaveh Da’at and Hazon Ovadia (Vol. 5), as well as most contemporary Poskim, prohibit both opening and using an umbrella on Shabbat. They contend that the case of the umbrella is critically different from the case of the folding chair. The open chair does not serve the space underneath it, whereas the open umbrella is designed to protect the person beneath it.
It is prohibited to open an umbrella on Shabbat and Yom Tob, as well as to use an already open umbrella.