"Zsuzsa's pub is closed, but her shop is open."
Translation:Zsuzsa kocsmája zárva van, de az üzlete nyitva.
When the possessor is a person's name, the definite article is frequently omitted. Not always though.
It is kind of similar to English. Try replacing "Zsuzsa" with something, eg. "the mayor". The definite article jumps right back in, in both languages.
Maybe it's my English failing me, but why is "van" in this sentence? I didn't think it was used after adjectives, and yet I swear "zárva" and "nyitva" are describing nouns here.