The Gemara (Berachot 32a) states: “One should always first state Hashem’s praises and only then begin to pray.” This mean that before praying to Hashem and requesting various things, it is proper to first speak words of praise to Hashem, as we find that Moshe Rabbeinu began by stating, “Hashem, G-d, you began to show your servant your greatness and your strong hand, for what god is like you in the Heavens or on the earth that can do according to your works and your mighty acts?” Only afterwards did Moshe request, “Let me please go over and see the good land that is beyond the Jordan.”
It is for this reason that the prevalent custom among the entire Jewish nation is to recite verses of song and praise to Hashem before beginning the actual prayer and this section is called “Pesukei De’Zimra”. Our Sages established that the “Baruch She’amar” blessing before beginning to recite these verses and the “Yishtabach” blessing at the conclusion of this section. Thus, these blessings were established for the Pesukei De’Zimra.
The Pesukei De’Zimra are considered a positive, time-bound Mitzvah, for these verses cannot be recited anytime one wishes and they may only be recited before the Shacharit prayer. As we have already established in a previous Halacha, the Shacharit prayer is limited, at the very latest, until halachic midday. Thus, the Pesukei De’Zimra may likewise not be recited past halachic midday. Indeed, Rav Natronai Gaon writes in one of his responses explicitly (quoted in the Siddur of Rav Amram Gaon, page 200) that Pesukei De’Zimra may not be recited after the Shacharit prayer. Maran Ha’Shulchan Aruch (Chapter 52) rules likewise.
The Mishnah (in the first Chapter of Masechet Kiddushin) states that although men are obligated to fulfill any positive, time-bound Mitzvah, women are exempt from such Mitzvot. Based on this, since Pesukei De’Zimra is a positive, time-bound Mitzvah, women are exempt from reciting it. Only righteous women who act piously recite these verses; however, they are not obligated to do so.
It is well-known that according to Maran Ha’Shulchan Aruch (Chapter 589), any Mitzvah that women are exempt from, even if they wish to perform it, they many nevertheless not recite a blessing upon it, for how can they recite the words “Asher Kideshanu Be’Mitzvotav Ve’Tzivanu” (Who has sanctified with his commandments and commanded us) when Hashem has never commanded them? Thus, women may not recite the blessings of “Baruch She’amar” and “Yishtabach” with Hashem’s name; rather, they must merely recite “Baruch Ha’el Ha’av Ha’Rachaman” and “Baruch Melech Mehulal Ba’Tishbachot”. Nevertheless, according to the Ashkenazi custom that women may recite a blessing even on Mitzvot that they are exempt from based on the ruling of the Rama (Chapter 17 and 589) who disagrees with the opinion of Maran Ha’Shulchan Aruch, women may recite these blessings of Pesukei De’Zimra as well (see Responsa Yechave Da’at, Volume 3, Chapter 3).
Summary: Women may not recite the “Baruch She’amar” and “Yishtabach” blessings with Hashem’s name. Nevertheless, the Ashkenazi custom is that even women may recite these blessings.