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If you're contemplating using one of these or any other service, investigate them first. PSA (Professional Sports Authenticator), for example, is a widely respected service, and even a cursory investigation will reveal this. Attaching a PSA certification to a book signed by a sports celebrity can add significant value to the book because they've built up widespread trust over time. Some authenticating services, frankly, are a joke. They serve more as a launch pad for fraud than they do as a defense against it.
The central problem with COA's is that anybody can issue one, and that includes forgers - in fact, a forger is more likely to. For this reason alone I've stopped using them because their very presence arouses suspicions of authenticity among many potential buyers. Instead, in cases where I feel it's necessary, I'll produce a COA-like document (including a context-defined photograph) that functions as a receipt of sale and includes both my signature and a guarantee of the signature for the life of the document. The difference is this: I call it a receipt. Even though my receipt still quacks like a duck, I think it's important not to call it a COA because to do so is tantamount to attaching an aura of "official" certification it simply doesn't possess.
Does Fran need a receipt? Probably not, but it's the work of a moment to insert a reassuring line in your listing, along with a money-back guarantee, or to include a much simpler version of a signed receipt of sale with the book. If it's Honest Abe, on the other hand (or anything else that may have significant value), he should get the 'A' treatment.
The question then becomes - your receipt or a full-blown COA from a third-party authenticator? The answer depends to some degree on who you are. If you have an established reputation in the business, your receipt will probably do in almost all cases. If not, or if there are circumstances surrounding the signature that compellingly call its authenticity into question, more work may need to be done.
Third-party authenticators are an option, of course, and significant fees will almost always apply, but a careful documentation of provenance, assuming it's available and substantive, can carry the day. Another option - and this is an especially good idea if you're preparing to sell the item - is to contact a major auction house with a dedicated books and manuscripts department. Three respected players here are Bonhams & Butterfields, Christie's, and Sotheby's. All have an online presence, and all will be able to both authenticate and appraise signed books. If you subsequently decide not to consign an item, however, fees may or may not apply. Ask.