We will give some new techniques for working with problems surrounding the Bourgain-Tzafriri Restricted Invertibility Theorem. First we show that the parameters which work in the theorem for all ‖T‖ ≤ 2 √ 2 closely approximate the parameters which work for all operators. This yields a generalization of the theorem which simultaneously does restricted invertibility on a small partition of the ve...

We examine some peculiarities of the subset of lattice preserving elements in a pseudo-Euclidean group, when the lattice under consideration contains a lightlike vector, or more generally, when the restriction of a pseudo-Euclidean metric to the real linear enveloppe of the lattice is not definite. For the case of a Lorentzian metric, it is shown in detail that the isometry group of the spaceti...

Let Ω be an arbitrary bounded domain of Rn. We study the right invertibility of the divergence on Ω in weighted Lebesgue and Sobolev spaces on Ω, and relate this invertibility to a geometric characterization of Ω and to weighted Poincaré inequalities on Ω. We recover, in particular, well-known results on the right invertibility of the divergence in Sobolev spaces when Ω is Lipschitz or, more ge...

The invertibility of second-order finite-difference operators with constant operator coefficients acting in the Banach space two-sided vector sequences is proved under condition their uniform injectivity (in particular, left invertibility), or surjectivity right Fredholm property.

The 1987 Bourgain-Tzafriri Restricted Invertibility Theorem is one of the most celebrated theorems in analysis. At the time of their work, the authors raised the question of a possible infinite dimensional version of the theorem. In this paper, we will give a quite general definition of restricted invertibility for operators on infinite dimensional Hilbert spaces based on the notion of density ...

The 1987 Bourgain-Tzafriri Restricted Invertibility Theorem is one of the most celebrated theorems in analysis. At the time of their work, the authors raised the question of a possible infinite dimensional version of the theorem. In this paper, we will give a quite general definition of restricted invertibility for operators on infinite dimensional Hilbert spaces based on the notion of density ...

In this article, we focus on the computation of statistics of invertible geometrical deformations (i.e., diffeomorphisms), based on the generalization to this type of data of the notion of principal logarithm. Remarkably, this logarithm is a simple 3D vector field, and is well-defined for diffeomorphisms close enough to the identity. This allows to perform vectorial statistics on diffeomorphism...

We investigate the invertibility of the morphological representation of binary images. A criteria for the invertibility of the morphological representation of binary images is proposed. Necessary and sufficient conditions for the exact reconstruction of a binary image from its morphological representation are provided.

5 Existing research provides no systematic, limited information procedure for handling 6 non-invertibility, despite the well-known inference problem it causes as well as its 7 presence in many types of dynamic systems. Non-invertibility means that structural 8 shocks cannot be recovered from a history of observed variables. It arises from a form 9 of delayed responses due to, among other things...