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Mohamed Zarrabi

Publications and source records attributed to Mohamed Zarrabi.

7 recordsLinked to original sources
arXiv2018

Explicit bounds for separation between Oseledets subspaces

We consider a two-sided sequence of bounded operators in a Banach space which are not necessarily injective and satisfy two properties (SVG) and (FI). The singular value gap (SVG) property says that two successive singular values of the cocycle at some index $d$ admit a uniform exponential gap; the fast invertibility (FI) property says that the cocycle is uniformly invertible on the fastest $d$-dimensional direction. We prove the existence of a uniform equivariant splitting of the Banach space into a fast space of dimension $d$ and a slow space of co-dimension $d$. We compute an explicit constant lower bound on the angle between these two spaces using solely the constants defining the properties (SVG) and (FI). We extend the results obtained in the finite-dimensional case for bijective operators and the results obtained by Blumenthal and Morris in the infinite-dimensional case for injective norm-continuous cocycles, in the direction that the operators are not required to be globally injective, that no dynamical system is involved, and no compactness of the underlying system or smoothness of the cocycle is required. Moreover, we give quantitative estimates of the angle between the fast and slow spaces that are new even in the case of finite-dimensional bijective operators in Hilbert spaces.

math.DS↗
arXiv2017

On powers of operators with spectrum in cantor sets and spectral synthesis

For $ξ\in \big( 0, \frac{1}{2} \big)$, let $E_ξ$ be the perfect symmetric set associated with $ξ$, that is $$E_ξ = \Big\{ \exp \Big( 2i π(1-ξ) \sum_{n = 1}^{+\infty} ε_{n} ξ^{n-1} \Big) : \, ε_{n} = 0 \textrm{ or } 1 \quad (n \geq 1) \Big\}$$ and $$b(ξ) = \frac{\log{\frac{1}ξ} - \log{2}}{2\log{\frac{1}ξ} - \log{2}}.$$ Let $q\geq 3$ be an integer and $s$ be a nonnegative real number. We show that any invertible operator $T$ on a Banach space with spectrum contained in $E_{1/q}$ that satisfies \begin{eqnarray*} & & \big\| T^{n} \big\| = O \big( n^{s} \big), \,n \rightarrow +\infty \\ & \textrm{and} & \big\| T^{-n} \big\| = O \big( e^{n^β} \big), \, n \rightarrow +\infty \textrm{ for some } β< b(1/q),\end{eqnarray*} also satisfies the stronger property $\big\| T^{-n} \big\| = O \big( n^{s} \big), \, n \rightarrow +\infty.$ We also show that this result is false for $E_ξ$ when $1/ξ$ is not a Pisot number and that the constant $b(1/q)$ is sharp. As a consequence we prove that, if $ω$ is a submulticative weight such that $ω(n)=(1+n)^s, \, (n \geq 0)$ and $C^{-1} (1+|n|)^s \leq ω(-n) \leq C e^{n^β},\, (n\geq 0)$, for some constants $C>0$ and $β< b( 1/q),$ then $E_{1/q}$ satisfies spectral synthesis in the Beurling algebra of all continuous functions $f$ on the unit circle $\mathbb{T}$ such that $\sum_{n = -\infty}^{+\infty} | \widehat{f}(n) | ω(n) < +\infty$.

math.FA↗
arXiv2017

A Szegö type theorem for truncated Toeplitz operators

Truncated Toeplitz operators are compressions of multiplication operators on $L^2$ to model spaces (that is, subspaces of $H^2$ which are invariant with respect to the backward shift). For this class of operators we prove certain Szegö type theorems concerning the asymptotics of their compressions to an increasing chain of finite dimensional model spaces.

math.FA↗
arXiv2010

Unitary equivalence to truncated Toeplitz operators

In this paper we investigate operators unitarily equivalent to truncated Toeplitz operators. We show that this class contains certain sums of tensor products of truncated Toeplitz operators. In particular, it contains arbitrary inflations of truncated Toeplitz operators; this answers a question posed by Cima, Garcia, Ross, and Wogen.

math.FA↗
arXiv2008

Compact operators that commute with a contraction

Let $T$ be a $C_0$--contraction on a separable Hilbert space. We assume that $I_H-T^*T$ is compact. For a function $f$ holomorphic in the unit disk $\DD$ and continuous on $\bar\DD$, we show that $f(T)$ is compact if and only if $f$ vanishes on $σ(T)\cap \TT$, where $σ(T)$ is the spectrum of $T$ and $\TT$ the unit circle. If $f$ is just a bounded holomorphic function on $\DD$ we prove that $f(T)$ is compact if and only if $\lim_{n\to \infty} T^nf(T) =0$.

math.FA↗