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Sektion Plenum
Mittwoch, 20.09.2000, 11.30 Uhr, Großer Mathematik-Hörsaal, Trefftz-Bau

Spektraleigenschaften von Blockoperatormatrizen und Anwendungen

Heinz Langer, Technische Universität Wien

Operators of the form

     (        )
        A  B
A  =    C  D
in the orthogonal sum H = H  o+ H of two Hilbert spaces H, H where the entries A, B, C, D are (in general unbounded) operators arise in many areas, e.g. in magnetohydrodynamics, systems and control theory, vibrating mechanical systems, quantum theory (Dirac and Klein-Gordon equation). Often the matrix A or its entries have certain symmetry or accretivity properties with respect to the inner product of the corresponding Hilbert spaces or with repect to some indefinite inner product on H. We give a survey of results concerning

the definition of the operator A in case of unbounded entries A, B, C, D,
the location of the spectrum of A ,
the dichotomy and the existence of special invariant subspaces of A ,
the completeness, partial and half range completeness of systems of root vectors of A ,
the block diagonalization of A ,
the existence of solutions of the corresponding Riccati equations.