Complete Second Order Linear Differential Equations in Hilbert Spaces

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Hilbert spaces are Banach spaces where the norm is induced by the inner product. Incomplete second order differential equations and first order equations in Banach spaces are a classical part of functional analysis. This text attempts to present a unified systematic theory of second order differential equations (y (t)+By’(t)+Cy(t)=0), including the well-posedness of the Cauchy, Dirichlet and Neumann problems; boundary conditions ensuring solvability of boundary-value problems; boundary behaviour and the extension of solutions on a finite interval; stabilization and stability of solutions at infinity; and boundary-value problems on a semi-line. The theory is developed in a special but important case, which can be considered as a model. Exhaustive answers to all the posed questions are given, with special emphasis on the effects arising for complete second order equations which do not arise for incomplete second order or first order equations. To achieve this, new results in the spectral theory of pairs of operators and the boundary behaviour of integral transformations have been developed.