Related papers: Shadowing, Generalized hyperbolic and Aluthge tran…
In this note, we verify that the bounded shadowing property and quasi-hyperbolicity of bounded linear operators on Hilbert spaces are preserved under Aluthge transforms.
It is rather well-known that hyperbolic operators have the shadowing property. In the setting of finite dimensional Banach spaces, having the shadowing property is equivalent to being hyperbolic. In 2018, Bernardes et al. constructed an…
We introduce and study the notions of (generalized) hyperbolicity, topological stability and (uniform) topological expansivity for operators on locally convex spaces. We prove that every generalized hyperbolic operator on a locally convex…
We introduce a class of linear bounded invertible operators on Banach spaces, called shift operators, which comprises weighted backward shifts and models finite products of weighted backward shifts and dissipative composition operators. We…
In the early 1970's Eisenberg and Hedlund investigated relationships between expansivity and spectrum of operators on Banach spaces. In this paper we establish relationships between notions of expansivity and hypercyclicity, supercyclicity,…
A linear operator $U$ acting boundedly on an infinite-dimensional separable complex Hilbert space $H$ is universal if every linear bounded operator acting on $H$ is similar to a scalar multiple of a restriction of $U$ to one of its…
We prove that if the Cauchy problem $\dot{u}=Au$ in a Banach space is hyperbolic, then the problem has the L-shadowing property. Conversely, if the space is finite-dimensional and the L-shadowing property is satisfied, then the problem is…
It is known that hyperbolic linear delay difference equations are shadowable on the half-line. In this paper, we prove the converse and hence the equivalence between hyperbolicity and the positive shadowing property for the following two…
A partially hyperbolic dynamical system is said to have the quasi-shadowing property if every pseudotrajectory can be shadowed by a sequence of points $(x_n)_{n\in \Z}$ such that $x_{n+1}$ is obtained from the image of $x_n$ by moving it by…
We describe the Aluthge transform of an unbounded weighted composition operator acting in an $L^2$-space. We show that its closure is also a weighted composition operator with the same symbol and a modified weight function. We investigate…
It is introduced an open class of linear operators on Banach and Hilbert spaces such that their non-wandering set is an infinite dimensional topologically mixing subspace. In certain cases, the non-wandering set coincides with the whole…
Our aim in this paper is to obtain necessary and sufficient conditions for weighted shift operators on the Hilbert spaces $\ell^{2}(\mathbb Z)$ and $\ell^{2}(\mathbb N)$ to be subspace-transitive, consequently, we show that the Herrero…
It is proved that a commutative algebra $A$ of operators in a reflexive real Banach space has an invariant subspace if each operator $T\in A$ satisfies the condition $$\|1- \varepsilon T^2\|_e \le 1 + o(\varepsilon) \text{ when }…
(Revised version, January 2006. S. Gouezel pointed out that, when 1<r<2, the proof in the previous version was incomplete. In fixing this gap, we simplified the argument in Section 6. In addition, there is a new appendix, with an…
We introduce the super-shadowing property in linear dynamics, where pseudotrajectories are approximated by sequences of the form $(\lambda_nT^nx)$, with $(\lambda_n)_n$ being complex scalars. For compact operators on Banach spaces, we…
A bounded operator on a real or complex separable infinite-dimensional Banach space $Z$ is universal in the sense of Glasner and Weiss if for every invertible ergodic measure-preserving transformation $T$ of a standard Lebesgue probability…
In this paper, we show that the generalized Aluthge transforma- tions of a large class of operators (weighted conditional type operators) are normal. As a consequence, the operator MwEMu is p-hyponormal if and only if it is normal, and…
Aluthge transform of a bounded operator is generalized to the case of unbounded one. A formula for the Aluthge transform of a weighted shift on a directed tree is established and it is used to construct an example of a hyponormal operator…
Let $T\in B(H)$ be a bounded linear operator on a Hilbert space $H$, let $T = V|T|$ be its polar decomposition of $T$ and let $\lambda\in [0,1]$. The $\lambda$-Aluthge transform $\Delta_{\lambda}(T)$ and the mean transforms $M(T)$ are…
It is proved that a commutative algebra $A$ of operators on a reflexive real Banach space has an invariant subspace if each operator $T\in A$ satisfies the condition $$\|1- \varepsilon T^2\|_e \le 1 + o(\varepsilon) \text{ when }…