### Browsing by Author "Tcaciuc, Adi"

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Item Controlling almost-invariant halfspaces in both real and complex settings(2017) Tcaciuc, Adi; Wallis, BenIf T is a bounded linear operator acting on an infinite-dimensional Banach space X, we say that a closed subspace Y of X of both infinite dimension and codimension is an almost-invariant halfspace (AIHS) under T whenever TY⊆Y+E for some finite-dimensional subspace E, or, equivalently, (T+F)Y⊆Y for some finite-rank perturbation F:X→X. We discuss the existence of AIHS’s for various restrictions on E and F when X is a complex Banach space. We also extend some of these and other results in the literature to the setting where X is a real Banach space instead of a complex one.Item The invariant subspace problem for rank one perturbations(2019) Tcaciuc, AdiWe show that for any bounded operator T acting on infinite dimensional, complex Banach space, and for any ε>0, there exists an operator F of rank at most one and norm smaller than ε such that T+F has an invariant subspace of infinite dimension and codimension. A version of this result was proved in \cite{T19} under additional spectral conditions for T or T∗. This solves in full generality the quantitative version of the invariant subspace problem for rank-one perturbations.Item Isometries of combinatorial Banach spaces(2020) Brech, C.; Ferenczi, V.; Tcaciuc, AdiWe prove that every isometry between two combinatorial spaces is determined by a permutation of the canonical unit basis combined with a change of signs. As a consequence, we show that in the case of Schreier spaces, all the isometries are given by a change of signs of the elements of the basis. Our results hold for both the real and the complex cases.Item On arithmetic progressions in model sets(2020) Klick, Anna; Strungaru, Nicolae; Tcaciuc, Adi; Strungaru, NicolaeIn this project we show the existence of arbitrary length arithmetic progressions in model sets and Meyer sets in the Euclidean d-space. We prove a van der Waerden type theorem for Meyer sets. We show that pure point subsets of Meyer sets with positive density and pure point diffraction contain arithmetic progressions of arbitrary length.Item On quasinilpotent operators and the invariant subspace problem(2019) Tcaciuc, AdiWe show that a bounded quasinilpotent operator T acting on an infinite dimensional Banach space has an invariant subspace if and only if there exists a rank one operator F and a scalar α∈ℂ, α≠0, α≠1, such that T+F and T+αF are also quasinilpotent. We also prove that for any fixed rank-one operator F, almost all perturbations T+αF have invariant subspaces of infinite dimension and codimension.