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Beiträge zu einer Perron-Frobenius-Theorie für Operatormatrizen. (Contribution to a Perron-Frobenius theory for operator matrices). (German) Zbl 0713.15008

Tübingen: Univ. Tübingen, Diss. 52 S. (1990).
The author considers unbounded operator matrices generating semigroups on products of Banach lattices. At first he presents a characterization of the positivity of the semigroup in terms of the generating operator matrix and its entries. Next he shows that by assuming order continuous Banach lattices there exists, under certain conditions, an explicit matrix representation for the generator of the minimal dominating semigroup (modulus semigroup). In the subsequent investigations a characterization of the irreducibility of positive semigroups generated by operators matrices is given (in particular, by the so-called companion matrices arising from higher order differential equations). Finally some results concerning the localization of the eigenvalues of a scalar matrix are extended to the case of operator matrices.
Reviewer: P.Charissiadis

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
47D03 Groups and semigroups of linear operators
15A18 Eigenvalues, singular values, and eigenvectors
15A30 Algebraic systems of matrices
47B60 Linear operators on ordered spaces
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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