## Regular Extensions of Hermitian OperatorsThe concept of regular extensions of an Hermitian (non-densely defined) operator was introduced by A. Kuzhel in 1980. This concept is a natural generalization of proper extensions of symmetric (densely defined) operators. The use of regular extensions enables one to study various classes of extensions of Hermitian operators without using the method of linear relations. The central question in this monograph is to what extent the Hermitian part of a linear operator determines its properties. Various properties are investigated and some applications of the theory are given. Chapter 1 deals with some results from operator theory and the theory of extensions. Chapter 2 is devoted to the investigation of regular extensions of Hermitian (symmetric) operators with certain restrictions. In chapter 3 regular extensions of Hermitian operators with the use of boundary-value spaces are investigated. In the final chapter, the results from chapters 1-3 are applied to the investigation of quasi-differential operators and models of zero-range potential with internal structure. |

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### Contents

Hermitian Operators | 9 |

Symmetric Operators | 18 |

Regular Extensions of Hermitian Operators | 26 |

Accretive Operators | 53 |

REGULAR EXTENSIONS WITH RESTRICTIONS | 61 |

SelfAdjoint NormPreserving Extensions of Hermitian Contractions | 67 |

Normal NormPreserving Extensions of Hermitian Contractions | 81 |

SelfAdjoint BoundPreserving Extensions of Semibounded Hermitian | 96 |

Description of Regular Extensions of Hermitian Operators | 146 |

Characteristic Functions of Hermitian Operators | 159 |

Spectral Properties of Regular Extensions | 166 |

EXAMPLES AND APPLICATIONS | 177 |

BoundaryValue Spaces for Model of ZeroRange Potentials | 189 |

Some Properties of Nonperturbed Operators | 201 |

Abstract Wave Equation | 215 |

Elements of the LaxPhillips Scattering Theory for pPerturbed | 234 |

Regular UInvariant Extensions of Hermitian Operators | 106 |

Canonical Dissipative Extensions of Hermitian Operators | 118 |

BOUNDARYVALUE SPACES OF HERMITIAN OPERATORS | 129 |

261 | |

### Common terms and phrases

abstract wave equation accretive operator arbitrary Assume ator Ax,x Ax,y bounded operator canonical dissipative extension Chapter characteristic function closed Hermitian operator conclude condition Consider the operator Corollary defect numbers defect subspace defined by decomposition defined by equality Denote densely defined dissipative operator element exists extension F extensions of Hermitian follows from equality Friedrichs extension Hence Hermitian contraction Hilbert space imply inequality Krein space Kuzhel Lemma Let us show lineal linear operator linearly independent maximal symmetric operator maximal uniformly positive Moreover Neumann formulas nonperturbed operator norm obtain operator A defined operator A0 operator acting operator H operator L orthonormal basis orthoprojector p-perturbed proof of Theorem regular extension scalar product scattering matrix self-adjoint extension self-adjoint operator semibounded semigroup simple maximal symmetric space H Subsection Theorem 2.1 translation representation U-invariant uniformly positive subspace unitary operator vector virtue of equalities