Since the discovery in the early 1950's, frames have emerged as an important tool in signal processing, image processing, data compression and sampling theory etc. Today, powerful tools from operator theory and Banach space theory are being introduced to the study of frames producing deep results in frame theory. In recent years, many mathematicians generalized the frame theory from Hilbert spaces to Hilbert C*-modules and got significant results which enrich the theory of frames. Also there is growing evidence that Hilbert C*-modules theory and the theory of wavelets and frames are tightly related to each other in many aspects. Both research fields can benefit from achievements of the other field. Our purpose of this dissertation is to work on several basic problems on frames for Hilbert C*-modules. We first give a very useful characterization of modular frames which is easy to be applied. Using this characterization we investigate the modular frames from the operator theory point of view. A condition under which the removal of element from a frame in Hilbert C*-modules leaves a frame or a non-frame set is also given. In contrast to the Hilbert space situation, Riesz bases of Hilbert C*-modules may possess infinitely many alternative duals due to the existence of zero-divisors and not every dual of a Riesz basis is again a Riesz basis. We will present several such examples showing that the duals of Riesz bases in Hilbert $C^*$-modules are much different and more complicated than the Hilbert space cases. A complete characterization of all the dual sequences for a Riesz basis, and a necessary and sufficient condition for a dual sequence of a Riesz basis to be a Riesz basis are also given. In the case that the underlying C*-algebra is a commutative W*-algebra, we prove that the set of the Parseval frame generators for a unitary group can be parameterized by the set of all the unitary operators in the double commutant of the unitary group. Similar result holds for the set of all the general frame generators where the unitary operators are replaced by invertible and adjointable operators. Consequently, the set of all the Parseval frame generators is path-connected. We also prove the existence and uniqueness of the best Parseval multi-frame approximations for multi-frame generators of unitary groups on Hilbert C*-modules when the underlying C*-algebra is commutative. For the dilation results of frames we show that a complete Parseval frame vector for a unitary group on Hilbert C*-module can be dilated to a complete wandering vector. For any dual frame pair in Hilbert C*-modules, we prove that the pair are orthogonal compressions of a Riesz basis and its canonical dual basis for some larger Hilbert C*-module. For the perturbation of frames and Riesz bases in Hilbert C*-modules we prove that the Casazza-Christensen general perturbation theorem for frames in Hilbert spaces remains valid in Hilbert C*-modules. In the Hilbert space setting, under the same perturbation condition, the perturbation of any Riesz basis remains a Riesz basis. However, this no longer holds for Riesz bases in Hilbert C*-modules. We also give a complete characterization on all the Riesz bases for Hilbert C*-modules such that the perturbation (under Casazza-Christensen's perturbation condition) of a Riesz basis still remains a Riesz basis.

Let A and B be real or complex C*-algebras. Let V and W be real or complex (right) full Hilbert C*-modules over A and B, respectively. Let T be a linear bijective map from V onto W. We show the following four statements are equivalent. (a) T is a unitary operator, i.e., there is a ∗-isomorphism £\ : A ¡÷ B such that <Tx,Ty> = £\(<x,y>), ∀ x,y∈ V ; (b) T preserves TRO products, i.e., T(x<y,z>) =Tx<Ty,Tz>, ∀ x,y,z in V ; (c) T is a 2-isometry; (d) T is a complete isometry. Moreover, if A and B are commutative, the four statements are also equivalent to (e) T is a isometry. On the other hand, if V and W are complex Hilbert C*-modules over complex C*-algebras, then T is unitary if and only if it is a module map, i.e., T(xa) = (Tx)£\(a), ∀ x ∈ V,a ∈ A.

In der vorliegenden Arbeit "Nichtkommutative Blochtheorie" beschäftigen wir uns mit der Spektraltheorie bestimmter Klassen von Hilbertraumoperatoren, den elliptischen Operatoren auf Darstellungsräumen von Hilbert-C*-Moduln. Die auftretenden C*-Algebren kodieren dabei Symmetrieeigenschaften der entsprechenden Operatoren.Für kommutative Symmetrien ist die Blochtheorie ein geeignetes Hilfsmittel. Wir schildern diese Methode zunächst in einem geometrischen Kontext, der allgemein genug ist, um die bekannten Ergebnisse über die Abwesenheit singulärstetigen Spektrums im Hinblick auf physikalische Anwendungen zu erweitern. Wir lassen uns dann durch eine Neuinterpretation der Blochtheorie aus einem nichtkommutativen Blickwinkel inspirieren zur Entwicklung einer nichtkommutativen Blochtheorie. Dabei werden bestimmte Eigenschaften von C*-Algebren verknüpft mit Eigenschaften des Spektrums elliptischer Operatoren. Diese Blochtheorie für Hilbert-C*-Moduln erlaubt es, verschiedene bekannte Resultate aus dem Bereich kommutativer (diskreter und kontinuierlicher) Geometrien mit nichtkommutativen Symmetrien in einem neuen gemeinsamen Rahmen zusammenzufassen, der Raum läßt für Modelle nichtkommutativer Geometrien mit nichtkommutativen Symmetrien. Wichtigstes Beispiel für die behandelte Klasse von Operatoren in der mathematischen Physik sind die Schrödingeroperatoren mit periodischem Magnetfeld und Potential. Wir ordnen sie in den Rahmen kommutativer und nichtkommutativer Blochtheorie ein und wenden die zuvor bereitgestellten Methoden an. / In this doctoral thesis "Nichtkommutative Blochtheorie'' (non-commutative Bloch theory) we investigate the spectral theory of a certain class of operators on Hilbert space: the elliptic operators associated with representations of Hilbert C*-modules. The C*-algebras that arise encode symmetry properties of the corresponding operators. For commutative symmetries Bloch theory is a proper tool. We describe this method in a geometric context which is general enough to extend known results about absence of singular continuous spectrum in view of physical applications. Then --- inspired by a new interpretation of Bloch theory from a non-commutative point of view --- we develop a non-commutative Bloch theory. Here certain properties of C*-algebras get linked to spectral properties of elliptic operators. This Bloch theory for Hilbert \CS-modules allows to unite, in a new common framework, several known results from the field of commutative (discrete and continuous) geometries having non-commutative symmetries; this leaves ample room for models of non-commutative geometries having non-commutative symmetries. In mathematical physics, the most important example for the class of operators considered is given by the Schrödinger operators with periodic magnetic field and potential. We place them into the framework of commutative and non-commutative Bloch theory and apply the methods developed before.

Positive kernels and their associated reproducing kernel Hilbert spaces have played a key role in the development of complex analysis and Hilbert-space operator theory, and they have recently been extended to the setting of free noncommutative function theory. In this paper, we develop the subject further in a number of directions. We give a characterization of completely positive noncommutative kernels in the setting of Hilbert C*-modules and Hilbert W*-modules. We prove an Arveson-type extension theorem for completely positive noncommutative kernels, and we show that a uniformly bounded noncommutative kernel can be decomposed into a linear combination of completely positive noncommutative kernels. / Ph. D.

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