Monotone convergence theorems for semi-bounded operators and forms with applications

Let H(n) be a monotone sequence of non-negative self-adjoint operators or relations in a Hilbert space. Then there exists a self-adjoint relation H(infinity) such that H(n) converges to H(infinity) in the strong resolvent; sense. This result and related limit results are explored in detail and new simple proofs are presented. The corresponding statements for monotone sequences of semi-bounded closed forms are established as immediate consequences. Applications and examples, illustrating the general results; include sequences of multiplication operators. Sturm-Lionville operators with increasing potentials, forms associated with Krein-Feller differential operators, singular perturbations of non-negative self-adjoint operators and the characterization of the Friedrichs and Krein-von Neumann extensions of a non-negative operator or relation.