A weighted bilateral shift with cyclic square is supercyclic

Shkarin, Stanislav

doi:10.1112/blms/bdm085

Cited by 5 publications

(3 citation statements)

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“…In [25], Shkarin proved the surprising result that a bilateral weighted shift operator T on 2 (Z) is supercyclic if and only if T 2 is cyclic! Using this fact together with Theorem 7.4 we get the following result.…”

Section: Thusmentioning

confidence: 98%

n-Weakly hypercyclic and n-weakly supercyclic operators

Feldman

2012

Journal of Functional Analysis

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If X is a locally convex topological vector space over a scalar field F = R or C and if E is a subset of X, then we define E to be n-weakly dense in X if for every onto continuous linear operator F : X → F n we have that F (E) is dense in F n . If X is a Hilbert space, this is equivalent to requiring that E have a dense orthogonal projection onto every subspace of dimension n. We then consider continuous linear operators on X that have orbits or scaled orbits that are n-weakly dense in X. We show that on a separable Hilbert space there are non-trivial examples of such operators and establish many of their basic properties. A fundamental tool is Ball's solution of the complex plank problem which implies that certain sets are 1-weakly closed.

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Section: Thusmentioning

confidence: 98%

n-Weakly hypercyclic and n-weakly supercyclic operators

Feldman

2012

Journal of Functional Analysis

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“…). Any such R admits both a dense of non-cyclic vectors, and a dense of cyclic vectors (see, e.g., [17,30]).…”

Section: Gain or Loss Of Krylov Solvability Under Perturbationsmentioning

confidence: 99%

Krylov solvability under perturbations of abstract inverse linear problems

Caruso¹,

Michelangeli²

2021

Preprint

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When a solution to an abstract inverse linear problem on Hilbert space is approximable by finite linear combinations of vectors from the cyclic subspace associated with the datum and with the linear operator of the problem, the solution is said to be a Krylov solution, i.e., it belongs to the Krylov subspace of the problem. Krylov solvability of the inverse problem allows for solution approximations that, in applications, correspond to the very efficient and popular Krylov subspace methods. We study here the possible behaviours of persistence, gain, or loss of Krylov solvability under suitable small perturbations of the inverse problem -the underlying motivations being the stability or instability of Krylov methods under small noise or uncertainties, as well as the possibility to decide a priori whether an inverse problem is Krylov solvable by investigating a potentially easier, perturbed problem. We present a whole scenario of occurrences in the first part of the work. In the second, we exploit the weak gap metric induced, in the sense of Hausdorff distance, by the Hilbert weak topology, in order to conveniently monitor the distance between perturbed and unperturbed Krylov subspaces.

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“…In [8], Menet generalized this result to the complex bilateral sequence spaces p (Z) with 1 ≤ p < ∞ and c•(Z) and afterwards, to the complex weighted spaces p (v, Z) and c•(v, Z). Shkarin, in [11] and [10] used the bilateral sequence spaces ∞(Z), p(Z) with 1 ≤ p < ∞ and c•(Z) to obtain various results associated with weighted bilateral shift on these spaces and also used {fj} j∈Z , a sequence of elements of B where B is a Banach space. We have introduced and studied the Banach space X−valued bilateral sequence spaces c•(Z, X, λ, p), c(Z, X, λ, p) in [15]…”

Section: Introductionmentioning

confidence: 99%