  We revise Krein's extension theory of positive symmetric operators. Our
approach using factorization through an auxiliary Hilbert space has several
advantages: it can be applied to non-densely defined transformations and it
works in both real and complex spaces. As an application of the results and the
construction we consider positive self-adjoint extensions of the modulus square
operator $T^*T$ of a densely defined linear transformation $T$ and bounded
self-adjoint extensions of a symmetric operator. Krein's results on the
uniqueness of positive (respectively, norm preserving) self-adjoint extensions
are also revised.
