M. Kreĭn’s Research on Semi-Bounded Operators, its Contemporary Developments, and Applications

TLDR: In this paper, the authors consider the M Kreĭn classical papers on semi-bounded operators and the theory of contractive self-adjoint extensions of Hermitian contractions, and discuss their impact and role in the solution of J von Neumann's problem about parametrization in terms of his formulas of all nonnegative selfadjoint extension of nonnegative symmetric operators.

Abstract: We are going to consider the M Kreĭn classical papers on the theory of semi-bounded operators and the theory of contractive self-adjoint extensions of Hermitian contractions, and discuss their impact and role in the solution of J von Neumann’s problem about parametrization in terms of his formulas of all nonnegative self-adjoint extensions of nonnegative symmetric operators, in the solution of the Phillips-Kato extension problems (in restricted sense) about existence and parametrization of all proper sectorial (accretive) extensions of nonnegative operators, in bi-extension theory of non-negative operators with the exit into triplets of Hilbert spaces, in the theory of singular perturbations of nonnegative self-adjoint operators, in general realization problems (in system theory) of Stieltjes matrix-valued functions, in Nevanlinna-Pick system interpolation in the class of sectorial Stieltjes functions, in conservative systems theory with accretive main Schrodinger operator, in the theory of semi-bounded symmetric and self-adjoint operators invariant with respect to some groups of transformations New developments and applications to the singular differential operators are discussed as well