  In 2002, Littlejohn and Wellman developed a celebrated general left-definite
theory for semi-bounded self-adjoint operators with many applications to
differential operators. The theory starts with a semi-bounded self-adjoint
operator and constructs a continuum of related Hilbert spaces and self-adjoint
operators that are intimately related with powers of the initial operator. The
development spurred a flurry of activity in the field that is still ongoing
today.
  The main goal of this expository (with the exception of Proposition 1)
manuscript is to compare and contrast the complementary theories of general
left-definite theory, the Birman--Krein--Vishik (BKV) theory of self-adjoint
extensions and singular perturbation theory. In this way, we hope to encourage
interest in left-definite theory as well as point out directions of potential
growth where the fields are interconnected. We include several related open
questions to further these goals.
