Is it permissible for a woman to say "O my beloved, O Omar" when mentioning one of the Companions?

Loving the Companions in general is an obligation upon every Muslim, and loving Umar (may Allah be pleased with him) in particular is also an obligation. There is no harm in a Muslim saying, when mentioning Umar or any other of the Companions, "O my beloved, so-and-so," for this is an expression of the love the heart harbors for this noble Companion, and "whoever loves a people will be resurrected with them."