118524160
Automatic Continuity of Linear Operators
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 [9], 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.
122589104
Bemerkungen zum von Neumannschen Minimaxtheorem
120206552
Beschränktheitsaussagen für sublineare und bisublineare Funktionale
122751787
Varianten zum Konvergenzsatz von Simons und Anwendungen in der Choquettheorie
120901930
Nonnegative jordan bases
In this paper we are concerned with the problem of when the generalized eigenspace of an n × n nonegative matrix A corresponding to its spectrai radius ρ has a nonnegative Jordan basis. Richman and Schneider have shown this to hold when, and only when, the Weyr characteristics of A corresponding to ρ is equal to the vector of levels in the singular graph of A Here we develop sufficient conditions for this equality to hold based upon the linear independence of a certain set of vectors which are generated according to a process of elimination of totally independent chains from the singular graph. We show that this condition comes close to being a further characterization for the existence of a nonnegative Jordan basis for this eigenspace and actually conjecture that this is the case. Finally, we show that this eigenspace has a Rothblum basis which is a Jordan basis if and only if the singular graph is a disjoint union of chains.
122950167
Monotonic sequences and rates of convergence of asynchronized iterative methods
Abstract In a recent paper B. Vemmer and the authors investigated the effect of varying the number of processors on the rate of convergence of the asynchronized parallel block Jacobi method associated with monotone matrices. It was found that, under certain simplifying assumptions, increasing the number of processors in relation to the number of blocks (or, what comes to the same in more general settings, the number of iteration operators) slows down the convergence. One interpretation for these results given in that paper was that increasing the number of processors means that when the current global approximation is updated by a local approximation from one of the processors, that local approximation was computed from a “much” earlier global approximation received from the host node. Hence the slowdown in the rate of convergence. The principal purpose of this paper is to remove some of the simplifying assumptions that were made in the above-mentioned paper and to prove that many of the results there hold under much more general conditions. Our present assumptions do not yield a fixed iteration matrix which models the process as was the case previously. This means that different tools have to be developed to establish results comparing the rate of convergence of two assynchronized processes.
121748192
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.
123146090
Corrections and additions to: "principal components of minus m-matrices"
121539345
On the sharpness of some upper bounds for the spectral radii of S.O.R. iteration matrices
SummarySharpness is shown for three upper bounds for the spectral radii of point S.O.R. iteration matrices resulting from the splitting (i) of a nonsingularH-matrixA into the ‘usual’D−L−U, and (ii) of an hermitian positive definite matrixA intoD−L−U, whereD is hermitian positive definite andL=1/2(A−D+S) withS some skew-hermitian matrix. The first upper bound (which is related to the splitting in (i)) is due to Kahan [6], Apostolatos and Kulisch [1] and Kulisch [7], while the remaining upper bounds (which are related to the splitting in (ii)) are due to Varga [11]. The considerations regarding the first bound yield an answer to a question which, in essence, was recently posed by Professor Ridgway Scott: What is the largest interval in ω, ω≧0, for which the point S.O.R. iterative method is convergent for all strictly diagonally dominant matrices of arbitrary order? The answer is, precisely, the interval (0, 1].