On the Strassen disintegration theorem

title: Automatic Continuity of Linear Operators
abstract: The present survey is essentially based on some recent automatic continuity results which have been obtained by Ernst Albrecht and the author in the joint papers [1;2;3]. In contrast to the excellent account of automatic continuity theory in Sinclair's lecture notes [29] and in Dales' recent survey article , the emphasis of the present exposition lies on the automatic continuity problem for linear operators acting between spaces of functions and distributions. The continuity of such operators will be derived from certain conditions which naturally arise in the theory of linear systems, namely from time-invariance and causality on the one side and from dissipativity on the other. The automatic continuity of all time-invariant and causal linear operators turns out to be just one achievement of a far-reaching new theory on automatic continuity which also applies to the problem of continuity for algebrahomomorphisms, derivations, intertwining operators, and local operators. Some central features of this theory will be sketched; in particular, some extensions and variants of the classical principle of uniform boundedness will be discussed.

title: Varianten zum Konvergenzsatz von Simons und Anwendungen in der Choquettheorie

title: Bemerkungen zum von Neumannschen Minimaxtheorem

title: Beschränktheitsaussagen für sublineare und bisublineare Funktionale

title: A maximality theorem concerning extreme points

title: On the spectrum of the SOR operator for symmetric positive definite matrices
abstract: One attractive feature of the classical successive overrelaxation (SOR) iterative method is its algorithmic simplicity. Whereas for linear systems with symmetric positive definite matrices A, the theorem of Ostrowski guarantees convergence as long as the relaxation parameter ω∈(0,2), results on the optimal choice of ω can only be obtained under certain additional assumtions on A. Here, for positive definite systems with coefficient matrices A=I−L−LT in which L is not necessarily strictly lower triangular, we study the behavior of the spectrum of the SOR operator Lω as ω→0 and ω→2 and describe enclosure sets which may be used to estimate the spectral radius of Lω for 0<ω<2. Numerical experiments show why results on the optimal relaxation parameter are difficult to obtain in the general case of positive definite systems.

title: Theorems of Stein-Rosenberg Type. III. The Singular Case*
abstract: In the theory of iterative methods, the classical Stein-Rosenberg theorem can be viewed as giving the simultaneous convergence (or divergence) of the extrapolated Jacobi (JOR) matrix J ω and the successive overrelaxation (SOR) matrix , in the case when the Jacobi matrix J 1 is nonnegative. As has been established recently by Buoni and Varga, necessary and sufficient conditions for the simultaneous convergence (or divergence) of J ω and have been established which do not depend on the assumption that J 1 is nonnegative. Our aim here is to extend these results to the singular case, using the notion of semiconvergence. In particular, for a real singular matrix A with nonpositive off-diagonal entries, we find conditions (Theorem 3.4) which imply that J ω and simultaneously semiconverge for all ω in the real interval [0,1).

title: Corrections and additions to: "principal components of minus m-matrices"

title: On the square roots of strictly quasiaccretive complex matrices
abstract: The square roots of a complex n × n matrix A for which the real part of e ix A , where χϵ[ −π 2 , π 2 ] , is positive definite are investigated. It is shown, for example, that when χ= −π 2 (i.e., A is strictly dissipative), A has a unique square root whose real and imaginary parts are both positive definite.

title: Some applications of partial orderings to iterative methods for rectangular linear systems
abstract: In this paper we extend the notions of K -semipositivity, K -monotonicity and of K -positive subinverses, introduced by Vandergraft for square matrices, to rectangular matrices. We apply these extensions to iterative methods for linear and rectangular systems Ax = b , A ϵ R m × n , induced by subproper and proper splittings of the matrix A .