You may have made a bit of a mistake by looking her up online. Don't stalk. And don't do anything that looks like stalking. But, the important thing, you may be able to graciously recover from this mistake. First off, don't look up anymore about her.
And then, pick a book about something you're interested in but that she is not, and this is the artful part. Go a little against type. For if she's somewhat interested and open to you, then she's going to be somewhat interested in what you're interested in, even if it's not one of her own.
And then the third time (or the fourth) may be just about the perfect time to ask her, "Would you like to grab a cup of coffee sometime?"
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Now, you may face a little bit of an uphill battle. For example, my seemingly 'nice' Harris County library, the librarians seem to stereotype regular users as potentially mentally ill using 1950s stereotypes, they really do. I wish this wasn't the case, but it sure seems to be the case. I even made a guess that 20% may be the threshold. If 20% of the regular users are in fact homeless or different, then they stereotype the whole group, and maybe this is one major avenue of how human prejudice develops.
There's nothing you can really do about this, except the sooner you ask her probably the better. (and esp don't do desk work and then half-pace, half-browse the rows and shelves as I sometimes do) And ask her in a brief, straightforward, winning, casual way and then one just never knows.