Extensions of derivations and symmetric operators

TL;DR

This paper generalizes the theory of boundary triples for symmetric and skew-symmetric operators, providing a parametrization of all their extensions using boundary quadruples and contractions, thus broadening the scope of extension theory.

Contribution

It introduces a new parametrization of all m-dissipative and selfadjoint extensions via boundary quadruples, even with differing defect indices, extending classical boundary triple theory.

Findings

Parametrization of m-dissipative extensions using contractions.

Characterization of unitary extensions generating C0-groups.

Unified framework for symmetric and skew-symmetric operator extensions.

Abstract

Given a densely defined skew-symmetric operators A 0 on a real or complex Hilbert space V , we parametrize all m-dissipative extensions in terms of contractions $\Phi$ : H-$\rightarrow$ H + , where Hand H + are Hilbert spaces associated with a boundary quadruple. Such an extension generates a unitary C 0-group if and only if $\Phi$ is a unitary operator. As corollary we obtain the parametrization of all selfadjoint extensions of a symmetric operator by unitary operators from Hto H +. Our results extend the theory of boundary triples initiated by von Neumann and developed by V. I. and M. L. Gorbachuk, J. Behrndt and M. Langer, S. A. Wegner and many others, in the sense that a boundary quadruple always exists (even if the defect indices are different in the symmetric case).