You don't say where you are located. Copyright laws are different in different countries, am going to assume US laws.
Under US law, a faithful digitization of a book does not get a new copyeight, see Bridgeman Art Library, Ltd. v. Corel Corp., 25 F. Supp. 2d 421 (S.D.N.Y. 1998) and thw Wikipedia article about that case
(On the issue of mrequired originality, see also FEIST PUBLICATIONS, INC. v. RURAL TELEPHONE SERVICE COMPANY, INC. (No. 89-1909.) (1991) which dealt with copying a telephone directory.)
The court in Bridgeman held that:
It is uncontested that Bridgeman's images are substantially exact
reproductions of public domain works, albeit in a different medium.
The images were copied from the underlying works without any avoidable
addition, alteration or transformation. Indeed, Bridgeman strives to
reproduce precisely those works of art.
...
The mere reproduction of
a work of art in a different medium should not constitute the required
originality for the reason that no one can claim to have independently
evolved any particular medium.'" As discussed above, the law requires
"some element of material alteration or embellishment" to the totality
of the work. At bottom, the totality of the work is the image itself,
and Bridgeman admittedly seeks to duplicate exactly the images of the
underlying works.
...
[O]ne need not deny the creativity inherent in the art of photography
to recognize that a photograph which is no more than a copy of the
work of another as exact as science and technology permit lacks
originality. That is not to say such a feat is trivial, simply not
original. The more persuasive analogy is that of a photocopier. Surely
designing the technology to produce exact reproductions of documents
required much engineering talent, but that does not make the
reproductions copyrightable.
The Bridgeman court was actually construing UK law, but the earthlier phase of Bridgeman i and the SCOTUS case of Feist show the same result under UIS law.
Note that books and other works published before 1925 are now out of copyright in the US. Copyright can also be lost ion other ways, such as publishing without a copyright notice before the effective date of the 1976 act, and failure to properly renew a work published in the US before 1964.
Assuming that the book is not under copyright, neither the library nor anyone else has a US copyright in the PFD. Unless the library imposes some additional restriction by contract, any such PDF may be copied or shared freely. It may even be sold or rented. And the validity of such an additional agreement would be questionable, but since the question does not mention such an agreement, I will not go into that further.
