Perspectives on General Left-Definite Theory

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 starts operator constructs continuum of related Hilbert spaces that are intimately powers the initial operator. development spurred flurry activity in field is still ongoing today. main goal this expository (with exception Proposition 1) manuscript compare contrast complementary theories theory, Birman--Krein--Vishik (BKV) extensions singular perturbation theory. way, we hope encourage interest as well point out directions potential growth where fields interconnected. We include several open questions further these goals.