Off-diagonal terms in symmetric operators (Q2737934)

Two obstructions to (essential) self-adjointness of a given symmetric operator \(S\) on some Hilbert space \({\mathcal H}\) are discussed. They are based on the classical von Neumann theory on deficiency indices and on the notion of smooth projections relative to \(S\), i.e. a family \((P_j)_{j\in\mathbb{N}}\) of projections in \({\mathcal H}\) such that the union of its ranges is dense in \({\mathcal H}\) and such that the range of each \(P_j\) is contained in the domain of \(S\). The results are then extended to finite families of symmetric operators which have a common dense and invariant domain in \({\mathcal H}\).