On the spectrum of the SOR operator for symmetric positive definite matrices
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: A combined direct-iterative approach for solving large scale singular and rectangular consistent systems of linear equations
abstract: We construct a certain iterative scheme for solving large scale consistent systems of linear equations Ax = b , (*) where A is a complex m X n matrix of rank r , m ⩾ n , and where A is assumed reasonably well conditioned. The iterative method is obtained through a careful exploitation of an LU -decomposition of A , and, disregarding roundoff errors, it converges to a solution to ( * ), though not necessarily the minimal l 2 -norm one, from any starting vector in r iterations. Moreover, once the LU -decomposition of A is complete, only about r 3 /2 arithmetic operations (multiplications and divisions) are needed to execute the r iterations.

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: A note on generalizations of strict diagonal dominance for real matrices
abstract: We investigate classes of real square matrices possessing some weakened from of strict diagonal dominance of a real matrix whose diagonal entries are all positive. The intersection of each one of these classes with the set of all real matrices, with nonpositive off-diagonal elements, coincides with the set of all nonsingular M - matrices.

title: Cone reachability for linear differential systems
abstract: Given a real nxn matrix A and a simplicial cone which is positively invariant under the linear differential system , we consider the problem of characterizing the closure of the set XA(K) of all initial points x(0) ∈ Rn which give rise to solutions x(t) that enter (and remain in) K. An explicit characterization of the closure of XA(K) is given first in case A is diagon-able with a real spectrum. Subsequently a characterization for the closure of XA(K) is given in case there exist k ∈ int K such that −Ak ∈ K, but in the absence of any other explicit assumptions on A.

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

title: Continuity of sublinear operators on F-spaces
abstract: In this note, some of the fundamental theorems concerning the automatic continuity of positive linear functionals and positive linear operators (see [9,14,21]) as well as certain uniform boundedness type theorems (see [5,13,15,19]) will be extended to derive various continuity properties of (even discontinuous) sublinear operators from some metrizable topological vector space to some ordered topological vector space.

title: Beschränktheitsaussagen für sublineare und bisublineare Funktionale

title: Bemerkungen zum von Neumannschen Minimaxtheorem

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.