Extensions of dissipative and symmetric operators

Abstract

AbstractGiven a densely defined skew-symmetric operator $$A_0$$ A 0 on a real or complex Hilbert space V, we parameterize all m-dissipative extensions in terms of contractions $$\Phi :{H_{{-}}}\rightarrow {H_{{+}}}$$ Φ : H - → H + , where $${H_{{-}}}$$ H - and $${H_{{+}}}$$ H + are Hilbert spaces associated with a boundary quadruple. Such an extension generates a unitary $$C_0$$ C 0 -group if and only if $$\Phi $$ Φ is a unitary operator. As a corollary we obtain the parametrization of all selfadjoint extensions of a symmetric operator by unitary operators from $${H_{{-}}}$$ H - to $${H_{{+}}}$$ 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).