You don't give full details, so it's hard to help.
That's a 1920 x 1080 display, so there shouldn't be an issue with the resolution, but the spec sheet says it was made in 2009, so maybe the age is an issue.
Anyways:
What year is your Air?
What MacOS are you running?
Are there any other types of input ports on that monitor that you could use instead and, if so, have you tried those?
Have you tried this with another computer to see if you get the same issue?
Also, please provide a screenshot of System Preferences > Displays showing your display settings. Is your computer detecting it as a 1920 x 1080 display?
You should also check out this thread, where people are experiencing problems with that monitor that sound similar to yours. It looks like their solution was to switch to the VGA input:
And according to this, you might try disabling auto-source
discussions.apple.com