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This dissertation is primarily concerned with studying the invariant subspaces of left-invertible, weighted shifts, with generalizations to left-invertible operators where applicable. The two main problems that are researched can be stated together as When does a weighted shift have the one-dimensional wandering subspace property for all of its closed, invariant subspaces? This can fail either by having a subspace that is not generated by its wandering subspace, or by having a subspace with an index greater than one. For the former we show that every left-invertible, weighted shift is similar to another weighted shift with a residual space, with respect to being generated by the wandering subspace, of dimension $n$, where $n$ is any finite number. For the latter we derive necessary and sufficient conditions for a pure, left-invertible operator with an index of one to have a closed, invariant subspace with an index greater than one. We use these conditions to show that if a closed, invariant subspace for an operator in a class of weighted shifts has a vector in $l^1$, then it must have an index equal to one, and to produce closed, invariant subspaces with an index of two for operators in another class of weighted shifts. / Ph. D.

This thesis consists of four chapters. Chapter 1 contains the preliminaries. We give the background, notation and some results needed for this work, and we describe our main results of this thesis. In Chapter 2 we will introduce a class of weighted Bergman spaces. We then will discuss some properties about the multiplication operator, Mz , on them. We also characterize the dual spaces of these weighted Bergman spaces. In Chapter 3 we will characterize the reducing subspaces of multiple weighted shifts. The reducing subspaces of the Bergman and the Dirichlet shift of multiplicity N are portrayed from this characterization. In Chapter 4 we will introduce the class of super-isometrically dilatable operators and describe their elementary properties. We then will discuss an equivalent description of the invariant subspace lattice for the Bergman shift. We will also discuss the interpolating sequences on the bidisk. Finally, we will examine a special class of super-isometrically dilatable operators. One corollary of this work is that we will prove that the compression of the Bergman shift on two compliments of two invariant subspaces are unitarily equivalent if and only if the two invariant subspaces are equal. / Ph. D.

Cette thèse s'articule autour de deux thèmes : une transformation de B(H) introduite par Aluthge et la méthode d'Ansari-Enflo. La première partie fait l'objet de l'étude de la transformation d’Aluthge qui a eu un impact important ces dernières années en théorie des opérateurs. Des résultats optimaux sur la stabilité d'un certain nombre de classes d'opérateurs, telles que la classe des isométries partielles et les classes associées au comportement asymptotique d'un opérateur, sont fournis. Nous étudions également l'évolution d'invariants opératoriels, tels que le polynôme minimal, la fonction minimum, l'ascente et la descente, sous l'action de la transformation ; nous comparons plus précisément les suites des noyaux et images relatives aux itérés d'un opérateur et de sa transformée de Aluthge. La deuxième partie est l'occasion d'étudier la théorie d'Ansari-Enflo, qui a permis de gros progrès pour le problème du sous-espace hyper-invariant. Nous développons plus particulièrement la notion fondatrice de la méthode, celle de vecteur extrémal. La localisation et une nouvelle caractérisation de ces vecteurs sont données. Leur régularité et leur robustesse, au regard de différents paramètres, sont éprouvées. Enfin, nous comparons les vecteurs extrémaux d'un shift à poids et ceux associés à sa transformée d’Aluthge. Cette étude aboutit à la construction d'une suite de vecteurs extrémaux associés aux itérés de la transformation d’Aluthge, pour laquelle certaines propriétés sont mises en évidence. / This thesis is based on two topics : a transformation of B(H) introduced by Aluthge and the Ansari-Enflo method. In the first part, we study the Aluthge transformation which really had an impact on operator theory in the past ten years. Some optimal results about stability for several operators classes, such as isometries class and classes of operators defined by their asymptotic behaviour, are given. We also study changes generated by Aluthge transform about some usual tools in operator theory like minimum polynomial, minimum function, ascent and descent ; precisely, we compare iterated kernels and iterated ranges sequences related to an operator and to its Aluthge transform. The second part is devoted to the study of the Ansari-Enflo theory, which allowed to make progress in the hyper-invariant subspace problem. We develop the notion of extremal vectors which is the fundamental point of the theory. We clarify their spatial localization and a new caracterisation for these vectors is given. Regularity and robustness with regard to different parameters are tried and tested. Finally, we compare extremal vectors associated with weighted shifts and the one corresponding to their Aluthge transform. This study leads to build a sequence of extremal vectors associated with the iterated Aluthge transform, for which we highlight several properties.