On generalized selfadjoint operators on scales of Hilbert spaces (Q2896634)

The notions of generalized selfadjoint and generalized essentially selfadjoint operators on a rigged Hilbert space were introduced by \textit{Yu. M. Berezansky} and \textit{J. Brasche} [Methods Funct. Anal. Topol. 8, No. 4, 1--14 (2002; Zbl 1074.47505)]. Such operators act from a positive Hilbert space to a negative space. The authors provide a series of examples (partial differential operators with constant coefficients, multiplication operators, differential operators with boundary conditions, all with the space \(L^2\) as basic space and the rigging by Sobolev spaces or weighted \(L^2\)-spaces), showing that a selfadjoint operator on a basic space can generate generalized essentially selfadjoint or nonselfadjoint operators. On the other hand, a generalized selfadjoint operator can correspond to a nonselfadjoint operator on \(L^2\).