### Browsing by Author "Tcaciuc, Adi"

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- ItemControlling 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.
- ItemIsometries 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.
- ItemOn 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.
- ItemOn 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.
- ItemThe 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.