This will be to collaborate with Isabelle Chalendar and Robert Eymard (LAMA/UGE) on a joint scientific research.
The Representation Theorem of Lions (RTL) is a version of the Lax–Milgram Theorem where completeness of
one of the spaces is not complete. The main point of our paper is a theory of derivations, based on RTL, for which well-posedness is proved. One application concerns non-autonomous evolution equations with a new initial-value and a periodic boundary condition for the time variable.
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 $\Phi:H_-\to H_+$, where $H_-$ and $H_+$ are Hilbert spaces associated with a boundary quadruple. 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 $H_-$ to $H_+$. Our results extend the theory of boundary triplets initiated by von Neumann and developed by V. I. and M .L. Gorbachuk, Wegner and many others.