If the library is a "connection" between the two schools, then it is implicitly part of the school property and your friend could most likely be banned from it--especially since 1) since schools are involved, the law tends to err on the side of safety--which would mean excluding someone from property used by a school, if that person had a disorderly offense; and 2) the library could almost certainly issue a banning notice directly, under the circumstances you describe. Your friend should use a different libary to avoid a legal fight he would likely lose.