Is it permissible to open an umbrella on Shabbat, or does this violate the prohibition against making an Ohel a tent-like structure on Shabbat? Needless to say, this question assumes that one does not carry the umbrella in an area where carrying is forbidden on Shabbat. Does opening the umbrella itself constitute a violation of Shabbat?
This issue generated considerable controversy among the later authorities, as discussed at length by the Bei'ur Halacha (supplementary work to the Mishna Berura by Rabbi Yisrael Meir Kagan, the "Chafetz Chayim," Lithuania, 1835-1933), in Siman 315. Those who allowed opening umbrellas on Shabbat equated umbrellas with a "Kisei Traskal," a kind of chair mentioned by the Gemara which resembled modern-day folding chairs. The Gemara ruled that one may open such a chair on Shabbat, even though he creates a "tent" over the area underneath the chair, because the chair is a single piece that requires no assembly, and one merely pulls the seat down to open the chair. Seemingly, this rationale should apply to an umbrella, as well, since opening an umbrella entails merely extending it outward, and not any kind of assembly or construction.
Others, however, distinguished between the case of a folding chair and that of umbrellas. The Gemara allowed opening a folding chair because one does not open it for the purpose of using the seat as an "Ohel" to protect him from the elements. An umbrella, by contrast, is intended for the specific purpose of protecting one from the rain or sun, and therefore indeed constitutes an "Ohel." Accordingly, the Noda Be-Yehuda (work of responsa by Rabbi Yechezkel Landau of Prague, 1713-1793), among others, forbade opening umbrellas on Shabbat. This is the position taken by the vast majority of recent and contemporary authorities, as well, including Rabbi Moshe Halevi, in his work Menuchat Ahava, and Chacham Ovadia Yosef, in his work Yechaveh Da'at. It should be noted that once Halacha forbids opening an umbrella, closing an umbrella becomes forbidden, as well.
Summary: One may not open or close an umbrella on Shabbat.