Die Dissertation untersucht die Invertierbarkeit im Unendlichen fuer Normgrenzwerte von Bandoperatoren - sogenannte band-dominierte Operatoren. Das dazu verwendete Instrument ist die Methode der Limitoperatoren. Es werden grundlegende Eigenschaften von Limitoperatoren bewiesen, Zusammenhaenge zur Invertierbarkeit im Unendlichen hergeleitet, sowie darueber hinaus gehende Anwendungen, z.B. zur Konvergenz von Projektionsverfahren, studiert.

This text is concerned with the Fredholm theory and stable approximation of bounded linear operators generated by a class of infinite matrices $(a_{ij})$ that are either banded or have certain decay properties as one goes away from the main diagonal. The operators are studied on $\ell^p$ spaces of functions $\Z^N\to X$, where $p\in[1,\infty]$, $N\in\N$ and $X$ is a complex Banach space. The latter means that our matrix entries $a_{ij}$ are indexed by multiindices $i,j\in\Z^N$ and that every $a_{ij}$ is itself a bounded linear operator on $X$. Our main focus lies on the case $p=\infty$, where new results are derived, and it is demonstrated in both general theory and concrete operator equations from mathematical physics how advantage can be taken of these new $p=\infty$ results in the general case $p\in[1,\infty]$.

This text is concerned with the Fredholm theory and stable approximation of bounded linear operators generated by a class of infinite matrices $(a_{ij})$ that are either banded or have certain decay properties as one goes away from the main diagonal. The operators are studied on $\ell^p$ spaces of functions $\Z^N\to X$, where $p\in[1,\infty]$, $N\in\N$ and $X$ is a complex Banach space. The latter means that our matrix entries $a_{ij}$ are indexed by multiindices $i,j\in\Z^N$ and that every $a_{ij}$ is itself a bounded linear operator on $X$. Our main focus lies on the case $p=\infty$, where new results are derived, and it is demonstrated in both general theory and concrete operator equations from mathematical physics how advantage can be taken of these new $p=\infty$ results in the general case $p\in[1,\infty]$.

Die Dissertation untersucht die Invertierbarkeit im Unendlichen fuer Normgrenzwerte von Bandoperatoren - sogenannte band-dominierte Operatoren. Das dazu verwendete Instrument ist die Methode der Limitoperatoren. Es werden grundlegende Eigenschaften von Limitoperatoren bewiesen, Zusammenhaenge zur Invertierbarkeit im Unendlichen hergeleitet, sowie darueber hinaus gehende Anwendungen, z.B. zur Konvergenz von Projektionsverfahren, studiert.

Dans cette thèse on prouve des résultats analytiques sur la théorie cohomotopique de Seiberg-Witten pour des 4-variétes Riemanniennes Spinc(4) a bouts périodiques, (X,g,τ). Nos résultats montrent, que sur certaines conditions techniques en (X, g, τ ),, cette nouvelle version est cohérente et mène a des invariants de Seiberg-Witten.Premièrement, en utilisant le critère de Taubes pour des operateurs périodiques dans des variétes a bouts périodiques, on montre que pour une 4-varieté Riemmanienne a bouts périodiques (X, g) vérifiant certaines conditions topologiques, le Laplacian ∆+ : L2(Λ2+) → L2(Λ2+) est un opérateur de Fredholm. On prouve une décomposition de type Hodge pour des 1-formes de X, a poids positif.Ensuite on prouve, en assumant certaines conditions topologiques et courbure scalaire non-negative sur les bouts, que l'opérateur de Dirac associé a une connection périodique (ASD a l'infini) est Fredholm.Dans la deuxième partie de la thèse on démontre un isomorphisme entre le groupe de cohomologie de de Rham Hd1R(X,iR), et le groupe harmonique intervenant dans la decomposition de Hodge des 1-formes de X a poids positif. On prouve l'existence de deux séquences exactes courtes liant le groupe de jauge de l'espace de modules de Seiberg-Witten et le groupe de cohomologie H1(X, 2πiZ).Dans la troisième partie on prouve les principaux résultats: la coercitivité de l'application de Seiberg-Witten et la compacité de l'espace de moduli pour une 4-varieté a bouts périodiques (X, g, τ ), vérifiant les conditions mentionnées plus haut.Finalment, utilisant la coercivité, on montre l'existence d'un invariant cohomotopique de type Seiberg- Witten type associé a (X, g, τ ). / In this thesis we prove analytic results about a cohomotopical Seiberg-Witten theory for a Riemannian, Spinc(4) 4-manifold with periodic ends, (X,g,τ) . Our results show that, under certain technical assumptions on (X, g, τ ), this new version is coher- ent and leads to Seiberg-Witten type invariants for this new class of 4-manifolds.First, using Taubes criteria for end-periodic operators on manifolds with periodic ends, we show that, for a Riemannian 4-manifold with periodic ends (X, g), verifying certain topological conditions, the Laplacian ∆+ : L2(Λ2+) → L2(Λ2+) is a Fredholm operator. This allows us to prove an important Hodge type decomposition for positively weighted Sobolev 1-forms on X.We prove, assuming non-negative scalar curvature on each end and certain technical topological conditions, that the associated Dirac operator associated with an end-periodic connection (which is ASD at infinity) is Fredholm.In the second part of the thesis we establish an isomorphism between be- tween the de Rham cohomology group, Hd1R(X,iR) (which is a topological in- variant of X) and the harmonic group intervening in the above Hodge type decomposition of the space of positively weighted 1-forms on X. We also prove two short exact sequences relating the gauge group of our Seiberg-Witten moduli problem and the cohomology group H1(X, 2πiZ).In the third part, we prove our main results: the coercivity of the Seiberg-Witten map and compactness of the moduli space for a 4-manifold with periodic ends (X,g,τ) verifying the above conditions.Finally, using our coercitivity property, we show that a Seiberg-Witten type cohomotopy invariant associated to (X, g, τ ) can be defined