The Strathprints institutional repository (https://strathprints.strath.ac.uk) is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output.On Abstract grad-div Systems.Rainer Picard, Stefan Seidler, Sascha Trostorff, Marcus WaurickAbstract. For a large class of dynamical problems from mathematical physics the skew-selfadjointness of a spatial operator of the form A = 0 −C * C 0 , where  is introduced (hence the title). As a particular application we consider a non-standard coupling mechanism and the incorporation of diffusive boundary conditions both modeled by setting associated with a skew-selfadjoint spatial operator A.

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In the mathematical analysis of scattering processes it is of particular interest to compare the long time behaviour of a given system with a more easily understood reference configuration. For applications it is important to have an abstract theory mimicking the naturally given assumptions of leading examples. This requirement is -with respect to an important class of problems related to selfadjoint realizations of partial differential operators -satisfied by the Birman-Belopolskii approach to two Hilbert space scattering theory, [Be]. The Birman-Belopolskii theory has experienced extension and considerable simplification by the work of Pearson, [Pe]. The occurence of infinite dimensional eigenspaces in examples of particular interest like Maxwelt's equations and the first order system oflinearized elasticity theory led to difficulties.To meet these difficulties Lyford proposed a more sophisticated setting of the assumptions [Ly 1], but his proof contained a gap, which was pointed out in [Ly 2]. Nevertheless, such a result is desirable.Fortunately we are able to introduce an additional assumption to Lyford's conditions, that seems to be quite naturally satisfied in the applications and that corrects the argument in Lyford's proof. We are concerned with the following situation.Let A~ be a selfadjoint operator with domain D(Ai) in the separable Hilbert space Hi, i = l, 2. In order to compare these operators we need an "identification operator"J: HI~H2 linear and bounded, J*(D(Az))CD(A1).Let (Hi(2))a~ ~ denote the spectral family of Ai and P~ be the projection on the corresponding absolute continuous subspace of H i, i= 1,2. The generalized wave operators W+ are the strong limits, when they exist, W e = s-lira W(t)Px

We consider a particular construction for skew-selfadjoint operator matrices, which are of central importance in initial boundary value problems of mathematical physics.Section 23: Applied operator theory Theorem 2.2 Let C generate an abstract grad-div system with C =