Operators with diskcyclic vectors subspaces

Abstract

In this paper, we prove that if T is diskcyclic operator then the closed unit disk multiplied by the union of the numerical range of all iterations of T is dense in H. Also, if T is diskcyclic operator and |λ| ≤ 1, then T − λI has dense range. Moreover, we prove that if α > 1, then 1 α T is hypercyclic in a separable Hilbert space H if and only if T ⊕ αI C is diskcyclic in H ⊕ C. We show at least in some cases a diskcyclic operator has an invariant, dense linear subspace or an infinite dimensional closed linear subspace, whose non-zero elements are diskcyclic vectors. However, we give some counterexamples to show that not always a diskcyclic operator has such a subspace.

Common diskcyclic vectors

“…Also, an operator 𝑇 is called diskcyclic if there is a vector 𝑥 ∈ 𝑋 called diskcyclic vector for 𝑇 such that the disk orbit 𝔻𝑂𝑟𝑏(𝑇, 𝑥) = {𝜆𝑇 𝑛 𝑥: 𝜆 ∈ ℂ, |𝜆| ≤ 1, 𝑛 ∈ ℕ} is dense in 𝑋 [3]. For more information on these operators, the reader may refer to [4][5][6].…”

$k$-bitransitive and compound operators on Banach spaces

“…A bounded linear operator T on a separable Banach space X is hypercyclic if there is a vector x ∈ X such that Orb(T, x) = {T n x : n ≥ 0} is dense in X, such a vector x is called hypercyclic for T , for more information on hypercyclic operators the reader may refer to [4,9]. Similarly, an operator T is called diskcyclic if there is a vector x ∈ X such that the disk orbit DOrb(T, x) = {αT n x : α ∈ C, |α| ≤ 1, n ∈ N} is dense in X, such a vector x is called diskcyclic for T , for more details on diskcyclicity see [2,3,12].…”