A Review of Some Works in the Theory of Diskcyclic Operators

Abstract

In this paper, we give a brief review concerning diskcyclic operators and then we provide some further characterizations of diskcyclic operators on separable Hilbert spaces. In particular, we show that if x ∈ H has a disk orbit under T that is somewhere dense in H then the disk orbit of x under T need not be everywhere dense in H. We also show that the inverse and the adjoint of a diskcyclic operator need not be diskcyclic. Moreover, we establish another diskcyclicity criterion and use it to find a necessary and sufficient condition for unilateral backward shifts that are diskcyclic operators. We show that a diskcyclic operator exists on a Hilbert space H over the field of complex numbers if and only if dim(H) = 1 or dim(H) = ∞ . Finally we give a sufficient condition for the somewhere density disk orbit to be everywhere dense.

On subspace-diskcyclicity

“…Suppose that N = X ⊕ {0}, and I is the identity operator on C 2 . Then, the operator S = T ⊕ I ∈ B(X ⊕ C 2 ) is not diskcyclic on X ⊕ X ; otherwise, we get I is diskcyclic operator on C 2 (see [2,Proposition 2.2]) which contradicts [2, Proposition 2.1]. However, it is clear that S is N -diskcyclic operator, and (x, 0) is N -diskcyclic vector for S. The next lemma gives some equivalent assertions to subspace-disk transitive, which will be the tool to prove several facts in this paper.…”

“…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 information about diskcyclic operators, the reader may refer to [2] [1] [12].…”

Characterization of Diskcyclic Operators

“…Later, in 2015 Bamerni, Kilicman and Noorani [1] presented a another equivalent version of the Diskcyclicity Criterion.…”

“…Theorem (Second Diskcyclicity Criterion). [1] Let T ∈ B(X). If there exists an increasing sequence of integers (n k ) k∈N and two dense sets…”

$k$-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. 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. In Banach spaces, hypercyclic (or diskcyclic) operators are identical to topological transitive (or disk transitive, respectively) [3,4]. 2. disk transitive, if for any two non empty open sets U and V, there exist a positive integer n and α ∈ C, 0 < |α| ≤ 1, such that T n αU ∩ V = ∅.…”