Resume followup
No matter what anyone says, a week is too soon to follow-up or update.
Times sure have changed. When I was actively seeking aviation employment ten years ago, the only ways to submit a resume were by mail, by personal hand-delivery, or by having someone walk it in to H.R. I sure don't like e-mailing a resume. It's too easy for a disinterested H.R. conehead to zap it. That puts you in the quandry of how to follow-up, because it is written that thou shalt be persistent but not annoy the pilot recruiter, lest he reacheth down from the heavens and blackball thee. I also don't like faxing resumes because H.R. is sure to be sick of facsimile spam. Also, the resume that you took so much time to design, write and polish won't show up as well on facsimile. Moreover, so many places scan resumes and a faxed resume may scan inaccurately.
Having said all that, the times to follow-up are when you add time and/or a rating, changed your status, e.g. new job or moved, or if you hear the place is hiring. Absent these conditions, update every four to six months.
Your resume should be accompanied by a good, focused and brief cover letter explaining the reason for the update. E.g. "I should like to update my file with the enclosed resume. I have added flight time." (paragraph) "I have flown an additional 200 hours since I submitted my first resume to you on June 1, 2002. My flight time now stands at 1000 total hours and 300 multiengine hours. I expect to add more time on my current flight schedule." (paragraph) I will appreciate any consideration for a Flight Officer position with Mediocre Commuter Airlines. Thank you again for your time." Something like that.
It depends on the company if you should call and ask if they received your resume. If it is a regional, you will annoy them. Regionals receive far too many resume to permit them to track each one, so calling to ask if a regional received your materials is sure to antagonize them. Other places might be different. Think about it carefully. I sure wouldn't send an e-mail under any circumstances because that kind of cold, impersonal, impudent communication is sure to antagonize.
Call me old school. I still think it's better to send in a resume by mail or hand-deliver it if possible. It's sure to be misplaced or ignored through other ways of transmittal.
Hope this helps. Good luck with your job search.