Do you perhaps have messages being filed automatically, perhaps to Junk or Spam? I see transient reports of incoming which is immediately moved to a folder by a filter.

Yes, I do, but I'm getting notifications with no incoming email in any folder or other account. I did notice, however, that there was something funky with the outgoing server setting for that account. It showed an empty field, and the pull-down didn't have all of the choices. I switched to the Outgoing Server (SMTP) entry in the overall account settings, and everything looked OK. When I went back to the settings for the problem account, the correct choice now appeared in the Outgoing Server (SMTP) field, and the notification problems seem to have disappeared.

Unfortunately, the problem is still there. I just received a notification of a new email on the problem account, checked the inbox and all other accounts and folders, and there's nothing new there. Checking the inbox cleared the notification, but it just came back again. This is annoying.