Proefschrift Universiteit van Amsterdam. / Met samenvatting in het Nederlands.

Let E and F be Hilbert C*-modules over a C*-algebra A. New classes of (possibly unbounded) operators t: E->F are introduced and investigated - first of all graph regular operators. Instead of the density of the domain D(t) we only assume that t is essentially defined, that is, D(t) has an trivial ortogonal complement. Then t has a well-defined adjoint. We call an essentially defined operator t graph regular if its graph G(t) is orthogonally complemented and orthogonally closed if G(t) coincides with its biorthogonal complement. A theory of these operators and related concepts is developed: polar decomposition, functional calculus. Various characterizations of graph regular operators are given: (a, a_*, b)-transform and bounded transform. A number of examples of graph regular operators are presented (on commutative C*-algebras, a fraction algebra related to the Weyl algebra, Toeplitz algebra, C*-algebra of the Heisenberg group). A new characterization of operators affiliated to a C*-algebra in terms of resolvents is given as well as a Kato-Rellich theorem for affiliated operators. The association relation is introduced and studied as a counter part of graph regularity for concrete C*-algebras.

Thesis (doctoral)--Technische Universiteit Eindhoven, 1989. / Includes indexes. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (p. 117-120).

Let E and F be Hilbert C*-modules over a C*-algebra A. New classes of (possibly unbounded) operators t: E->F are introduced and investigated - first of all graph regular operators. Instead of the density of the domain D(t) we only assume that t is essentially defined, that is, D(t) has an trivial ortogonal complement. Then t has a well-defined adjoint. We call an essentially defined operator t graph regular if its graph G(t) is orthogonally complemented and orthogonally closed if G(t) coincides with its biorthogonal complement. A theory of these operators and related concepts is developed: polar decomposition, functional calculus. Various characterizations of graph regular operators are given: (a, a_*, b)-transform and bounded transform. A number of examples of graph regular operators are presented (on commutative C*-algebras, a fraction algebra related to the Weyl algebra, Toeplitz algebra, C*-algebra of the Heisenberg group). A new characterization of operators affiliated to a C*-algebra in terms of resolvents is given as well as a Kato-Rellich theorem for affiliated operators. The association relation is introduced and studied as a counter part of graph regularity for concrete C*-algebras.:Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Sightings 1. Unitary *-module spaces Algebraic essence of adjointability on Hilbert C*-modules . . . . . 13 a) Operators on Hilbert C*-modules - Notions. . . . . . . . . . . . . . 13 b) Essential submodules and adjointability . . . . . . . . . . . . . . . . 15 c) From Hilbert C*-modules to unitary *-module spaces . . . . . . 16 2. Operators on unitary *-module spaces Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3. Graph regularity Pragmatism between weak and (strong) regularity . . . . . . . . . 27 a) Types of regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 b) The case C(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 c) Graph regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Transition. Orthogonal complementability and topology Back to Hilbert C*-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Graph regular operators on Hilbert C*-modules 4. Commutative case: Operators on C_0(X) Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Interjection. Unboundedness and graph regularity . . . . . . . . . . 55 5. Relation to adjointable operators Sources of graph regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6. Concrete C*-algebras Association relation and affiliation relation . . . . . . . . . . . . . . . . 61 7. Examples Graph regular operators that are not regular . . . . . . . . . . . . . 67 a) Position and momentum operators as graph regular operators on a fraction algebra related to the Weyl algebra . . 67 b) A graph regular but not regular operator on the group C*-algebra of the Heisenberg group . . . . . . . . . . . . . . . 69 c) Unbounded Toeplitz operators . . . . . . . . . . . . . . . . . . . . . . . 70 8. Bounded transform The canonical regular operator associated to a graph regular operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 9. Absolute value and polar decomposition . . . . . . . . . . . . . . . 79 10. Functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 11. Special matrices of C*-algebras Counter examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Abstract and open questions . . . . . . . . . . . . . . . . . . . . . . . . . 89 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Dank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Erklärung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Die vorliegende Arbeit beschäftigt sich mit der Bures-Geometrie auf Zustandsräumen über von-Neumann-Algebren. Diese basiert auf jenem Abstandsbegriff für normale Zustände, der von Bures im Jahre 1969 eingeführt wurde. Eng damit verbunden ist der Begriff der algebraischen Übergangswahrscheinlichkeit, der von Uhlmann 1976 vorgeschlagen wurde. An einem Beispiel wird gezeigt, dass man den Bures-Abstand unter Umständen nicht implementieren kann, wenn man einen der implementierenden Vektoren vorgeben will. Im weiteren wird der vom Bures-Abstand induzierte Paralleltransport von Vektoren entlang Loops von normalen Zuständen untersucht. Um die Holonomiegruppe im unendlichdimensionalen Fall zu untersuchen, werden Sätze über Produkte positiver Operatoren hergeleitet. Diese Sätze, die durchaus auch von eigenständigem Interesse sein könnten, werden mit Ergebnissen aus der Literatur verglichen. Schließlich wird der Bures-Abstand unter infinitesimalem Blickwinkel betrachtet. Die so entstehenden Bures-geodätischen Bögen werden untersucht. Speziell wird gefragt, ob gewisse Strata stets geodätisch konvex sind, also als Beispiel für Umgebungen dienen können. Um diese Frage am Ende negativ zu beantworten, werden mehrere Sätze über Sakaische Radon-Nikodym-Operatoren hergeleitet, die auch ohne Bezug zur Bures-Geometrie interessant sein könnten. Das entscheidende Gegenbeispiel nutzt Gohbergs Ergebnis zum Spektrum bestimmter Toeplitzoperatoren aus. Ein Nebeneffekt des beschriebenen Verfahrens ist, dass es auch zur Konstruktion von Operatoren mit hinreichend nichttrivialem Spektrum benutzt werden kann. / The present paper deals with Bures' geometry in the state space over von-Neumann algebras. This geometry is based on the distance introduced by Bures in 1969. Closely related with it is the concept of algebraic transition probability as proposed by Uhlmann in 1976. It is shown by an example that there are cases where one can not implement Bures' distance if one of the implementing vectors is given. In the following, the parallel transport of vectors along loops of normal states, which is induced by Bures' distance, is examined. In order to investigate the holonomy group in the infinite-dimensional case, theorems on products of positive operators are derived. These theorems, which could be of interest on their own, are compared with the literature. Finally, Bures' distance is examined infinitesimally. The thus arising Bures-geodesic arcs are investigated. Especially, it is asked whether certain strata are geodesically convex and can therefore serve as examples of neighbourhoods. In order to finally give a negative answer, several theorems on Sakai's Radon-Nikodym operators, which could also be of interest without a connection to Bures' geometry, are derived. The critical counterexample exploits Gohberg's result on the spectrum of certain Toeplitz operators. A by-product of the described procedure is that it can be used to construct operators which have a sufficiently non-trivial spectrum.

The overreaching goal of this paper is to investigate the existence and uniqueness of weak solution of a semilinear parabolic equation with double nonlocality in space and in time variables that naturally arises while modeling a biological nano-sensor in the chaotic dynamics of a polymer chain. In fact, the problem under consideration involves a symmetric integrodifferential operator of Lévy type and a term called the interaction potential, that depends on the time-integral of the solution over the entire interval of solving the problem. Owing to the Galerkin approximation, the existence and uniqueness of a weak solution of the nonlocal complement value problem is proven for small time under fair conditions on the interaction potential.

Ziel dieser Arbeit ist die Erarbeitung von effizienten (semi-)automatischen Verfahren zur Verknüpfung von Wissensbasen. Eine Vielzahl von Lösungsklassen können zu diesem Zweck eingesetzt werden. In dieser Arbeit werden ausschließlich deklarative Ansätze erörtert. Deklarative Ansätze gehen davon aus, dass das direkte Errechnen von Mappings zwischen Mengen von Ressourcen in vielen Fällen nur schwer möglich ist oder eines nicht vertretbaren Aufwands bedarf. Diese Ansätze zielen daher darauf ab, eine Ähnlichkeitsfunktion sowie einen Schwellwert zu finden, die zur Approximation eines Mappings genutzt werden können. Zwei Herausforderungen gehen mit dieser Modellierung des Problems einher: (a) Effizienz sowie (b) Genauigkeit und Vollständigkeit. Lösungen zu beiden Herausforderungen sowie auf echten Daten basierende Anwendungen dieser Lösungen werden in der Arbeit vorgestellt.

We study a class of differential inclusions involving maximal monotone relations, which cover a huge class of problems in mathematical physics. For this purpose we introduce the time derivative as a continuously invertible operator in a suitable Hilbert space. It turns out that this realization is a strictly monotone operator and thus, the question on existence and uniqueness can be answered by well-known results in the theory of maximal monotone relations. Furthermore, we show that the resulting solution operator is Lipschitz-continuous and causal, which is a natural property of evolutionary processes. Finally, the results are applied to a system of partial differential equations and inclusions, which describes the diffusion of a compressible fluid through a saturated, porous, plastically deforming media, where certain hysteresis phenomena are modeled by maximal montone relations.

A wealth of multidimensional OLAP models has been suggested in the past, tackling various problems of modeling multidimensional data. However, all of these models focus on navigational and query operators for grouping, selection and aggregation. We argue that planning functionality is, next to reporting and analysis, an important part of OLAP in many businesses and as such should be represented as part of a multidimensional model. Navigational operators are not enough for planning, instead new factual data is created or existing data is changed. To our knowledge we are the first to suggest a multidimensional model with support for planning. Because the main data entities of a typical multidimensional model are used both by planning and reporting, we concentrate on the extension of an existing model, where we add a set of novel operators that support an extensive set of typical planning functions.

No description available.