Although this may seem obvious, remember that taking no action on a customer complaint doesn't mean you don't need to respond to the customer. I have had many conversations with people who think that, because we couldn't find the root cause, or there wasn't enough information from the customer, etc., that we didn't even have to acknowledge the complaint.
The best thing is to open dialogue with the customer and let them know what you are doing, what you have found. The more they know about your process of investigation, the more likely you are to get assistance from them in closing it, or even the benefit of the doubt in the future when they are thinking of issuing a complaint.