The paper deals with the generalized Drazin invertibility of combinations of idempotents p, q in a Banach algebra. It proves the equivalence of the generalized Drazin invertibility of p−q and p+q, as well as the equivalence of the generalized Drazin invertibility of the commutator pq − qp and anticommutator pq + qp of p, q. It extends several results of J. Math. Anal. Appl. 359 (2009) 731–738, ...

We review the concepts of local and global invertibility for a nonlinear auto-regressive moving-average (NLARMA) model. Under very general conditions, a local invertibility analysis of a NLARMA model admits the generic dichotomy that the innovation reconstruction errors either diminish geometrically fast or grow geometrically fast. We derive a simple sufficient condition for a NLARMA model to b...

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...

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 ...

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.