Resumen de A continuous model for quasinilpotent operators and it connection with translation invariant subspaces in weighted l2 on r+
Daniel José Rodríguez Luis
The aim of this monograph has been to provide a better understanding for the class of quasinilpotent operators in the context of invariant subspaces. This monograph is organized as follows.
In Chapter 1 we present the notation and preliminary results that will be useful in the following chapters. In the first part, we recall the concept of a quasitriangular operators introduced by Halmos in order to solve the Invariant Subspace Problem. In addition, we introduce the notion of a model for bounded linear operators due to Rota and show some important results in the literature. Finally, we state, but not in full detail, a discrete model for quasinilpotent operators due to Foias and Pearcy. In the second part, we introduce the problem of translation invariant subspaces for the classical space of functions $L^2$ on different groups such as $Z$ and $R$. We show the well-known Beurling-Lax Theorem and recall the concept of unicellular operators, which is a crucial concept in Chapter 3 and Chapter 4.
In Chapter 2 we prove a continuous version of the model due to Foias and Pearcy for quasinilpotent operators defined on a separable, infinite-dimensional complex Hilbert space, providing a new approach for studying their invariant subspaces. This continuous version asserts that every quasinilpotent operator defined on a separable, infinite-dimensional complex Hilbert space can be viewed as the restriction of the backward shift operator $(S^*f)(t)=f(t+1)$, $t\geq0$ to one of its invariant subspaces $\M$ when $S^*$ is defined in $L^2(R_+,w(t)dt)$, for some positive function $\alpha\in C^\infty(R_+)$.
In Chapter 3 we provide an extension of a theorem of Domar showing that the ordered lattice of invariant subspaces for the semigroup of right shift $\{S_\tau\}_{\tau\geq0}$ in $L^2(R_+,w(t)dt)$ may be obtained for a wider class of positive functions not fulfilling the log-concavity assumption on the weight $w$. In addition, we show examples of positive continuous functions in the context of our theorem but not fulfilling the log-concavity assumption in strong sense.
Finally, in Chapter 4 we show that the uniformly bounded jump condition is, in some sense, crucial in order to obtain an ordered lattice of invariant subspaces. This result is a continuous version of a theorem due to Nikolskii in the context of weighted shift operators in the sequence space $\ell^2(Z_+)$. In this sense, we prove the existence of a large class of positive continuous functions $\beta$ such that the lattice of the semigroup $\{S_\tau\}_{\tau\geq0}$ in $L^2(R_+,\beta(t)dt)$ is not ordered by inclusion.
