To resolve is used differently than solve. To resolve is used when there are (usually)two differing opinions and there needs to be a decision made between the two. As rogelio says, resolve is used mainly with an argument between two people.
Resolve is also used as a noun. "He has a lot of resolve", meaning: you have a lot of determination or you have the ability to put your mind to something deliberately.
To solve a problem means to find an answer using all the information provided and determining the answer. To solve the crime, the police used all the evidence they found in the apartment.
to me, resolve gives the impression of a compromise. like jacinta said, if you have two differing opinions, and you resolve the argument, that could mean that the two people compromised rather than one or the other winning. There could be many ways to resolve an argument.
so to resolve something you need to have some sort of conflicting options in the first place.
interestingly, I wouldn't tend to use "resolution" in this situation. If my friend and I resolved our argument, I would still say that we found a solution to the argument! I don't know why, maybe because these days "resolution" is used more as a computer term?
I was thinking of using "resolve" in a sentence like this: In order to try to find an absolute measure of linguistic simplicity, we could look into sound systems and resolve that the fewer letters and the simpler syllabic structure a language has, the simpler its sound system is. Could I use resolve there? If not, which other verb would you recommend?
"Resolved" means that the problem has been fixed. The questions is what what 'should be' means in this context. It could mean that the writer thinks that the problem has been resolved, based on what they know --- not because they have tried and seen that the problem is gone. That's what I think it probably means. However, it could mean that someone should fix the problem.
The context should tell you which is more likely. Which one do you think fits best?
Thanks Cagey, Yes, it's all about the context.
The actual announcement - Apps Status Dashboard
I can just conclude that language is interesting. Say, think about the same sentence ("This should be resolved.") said by a Gmail management staff to a troubleshooting engineer, or from Gmail to its users. Interesting.