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 $\overline{\mathrm{d}\mathrm{o}\mathrm{m}}S$ are studied. The main focus is on the case $\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}\overline{\mathrm{d}\mathrm{o}\mathrm{m}}S=\mathrm{\infty }$. New properties of the characteristic functions of non-densely defined symmetric operators are established.