Compressions of selfadjoint and maximal dissipative extensions of non-densely defined symmetric operators

Selfadjoint and maximal dissipative extensions of a non-densely defined symmetric operator $S$ in an infinite-dimensional separable Hilbert space are considered and their compressions on the subspace ${\rm \overline{dom}\,} S$ are studied. The main focus is on the case ${\rm codim\,}{\rm \overline{dom}\,}S=\infty$. New properties of the characteristic functions of non-densely defined symmetric operators are established.