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Adi Tcaciuc

Publications and source records attributed to Adi Tcaciuc.

10 recordsLinked to original sources
arXiv2020

Invariant subspace problem for rank-one perturbations: the quantitative version

We show that for any bounded operator $T$ acting on infinite dimensional, complex Banach space, and for any $\varepsilon>0$, there exists an operator $F$ of rank at most one and norm smaller than $\varepsilon$ 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.

math.FA↗
arXiv2020

On arithmetic progressions in model sets

In 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.

math.DS↗
arXiv2019

On quasinilpotent operators and the invariant subspace problem

We 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 $α\in\mathbb{C}$, $α\neq 0$, $α\neq 1$, 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.

math.FA↗
arXiv2018

The invariant subspace problem for rank one perturbations

We show that for any bounded operator $T$ acting on an infinite dimensional Banach space there exists an operator $F$ of rank at most one such that $T+F$ has an invariant subspace of infinite dimension and codimension. We also show that whenever the boundary of the spectrum of $T$ or $T^*$ does not consist entirely of eigenvalues, we can find such rank one perturbations that have arbitrarily small norm. When this spectral condition is not satisfied, we can still find suitable finite rank perturbations of arbitrarily small norm, but not necessarily of rank one.

math.FA↗
arXiv2016

Controlling almost-invariant halfspaces in both real and complex settings

If $T$ 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 $TY\subseteq Y+E$ for some finite-dimensional subspace $E$, or, equivalently, $(T+F)Y\subseteq Y$ for some finite-rank perturbation $F:X\to X$. 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.

math.FA↗
arXiv2012

Every operator has almost-invariant subspaces

We show that any bounded operator $T$ on a separable, reflexive, infinite-dimensional Banach space $X$ admits a rank one perturbation which has an invariant subspace of infinite dimension and codimension. In the non-reflexive spaces, we show that the same is true for operators which have non-eigenvalues in the boundary of their spectrum. In the Hilbert space, our methods produce perturbations that are also small in norm, improving on an old result of Brown and Pearcy.

math.FA↗
arXiv2009

Almost invariant half-spaces of operators on Banach spaces

We introduce and study the following modified version of the Invariant Subspace Problem: whether every operator T on a Banach space has an almost invariant half-space, that is, a subspace Y of infinite dimension and infinite codimension such that Y is of finite codimension in T(Y). We solve this problem in the affirmative for a large class of operators which includes quasinilpotent weighted shift operators on l_p (1 \le p < \infty) or c_0.

math.FA↗