## Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 927-951 |

Number of pages | 25 |

Journal | Proceedings of the royal society of edinburgh section a-Mathematics |

Volume | 140 |

Publication status | Published - 2010 |

## Keywords

- SELF-ADJOINT OPERATORS
- RANK-ONE PERTURBATIONS
- NONNEGATIVE OPERATORS
- EXTENSION THEORY