On Subspace Codisk-Cyclicity

. Let 𝒩 be a subspace of an infinite dimensional complex separable on a Hilbert space ℋ . The operator 𝑇 ∈ ℬ(ℋ) is said to be 𝒩 -codisk-cyclic, if there is a nonzero vector 𝑦 in ℋ , then 𝒩 ∩ {𝛽𝑇 𝑛 𝑦: 𝛽 ∈ ℂ, |𝛽| ≥ 1, 𝑛 ∈ ℕ} is dense in 𝒩 . This paper, introduces the properties of the concepts 𝒩 -codisk-cyclic and 𝒩 -codisk-cyclic transitive. The existence of a subspace codisk-cyclic operator on 𝑛 -dimensional complex Hilbert space is illustrated and a criterion of 𝒩 -codisk-cyclic operator in infinite dimensional is obtained.


INTRODUCTION
Let ℬ(ℋ) be the algebra of all bounded linear operators on a separable infinite dimensional Hilbert space ℋ, ∈ ℬ(ℋ) is said to be hypercyclic operator, if the orbit of with a nonzero vector in ℋ, then orbt( , ) ≔ { : ≥ 0} is dense in ℋ. Thus is said to be a hypercyclic vector for [1] ISSN: 0067-2904 Jamil and Hamada Iraqi Journal of Science, 2023, Vol. 64, No. 6, pp: 3993-3997 3994 The motivation of the studying of the scalar multiples of an orbit is due to the example of Rolewicz [2]. The operator is called a supercyclic operator, if there exists a nonzero vector where the cone generated by ( , ) is dense in ℋ its definition is created by Hilden and Wallen in 1974 [3] Hypercyclicity are extensively studied by many researchers, for more detail see [1] [4].
In 2010, Jamil in [8] shown that, if the ( , ) is somewhere dense, then it is everywhere dense, that is closure of ( ( , )) ≠ ∅, then must be codisk-cyclic. Hence, to discuss codisk-cyclicity for closed sets , it must have empty interior, e.g., is a nontrivial subspace.
The concepts of subspaces codisk-cyclicity and codisktransitivity are presented in this paper. We give an example to ensure that not every subspace codisk-cyclic operator is codiskcyclic. Some necessary and sufficient conditions of subspace codisk-transitive operators are investigated. Moreover, a subspace codisk-cyclic criterion is established, and discussed when these two concepts (subspaces codisk-cyclicity and codisktransitivity) are equivalence. We will abbreviate the set { ∈ ℂ: | | ≥ 1} by , { ∈ ℂ: | | ≤ 1} by and ℕ ⋃{0} by ℕ 0 .

Subspace Codisk-Cyclic
In this section, we introduce a subspace codisk-cyclic operator and study some of its properties.

Subspace Codisk-Cyclic transitive
In this section, we aim to introduce the subspace Codiskcyclic transitive and study some of its properties.

3) For all non-empty relatively open sets
, ⊆ , there exist ∈ ℕ 0 and ∈ , such that ∩ ( ) is a non-empty relatively open set of . Now we turn our attention to discuss the necessary condition for an operator to be −codisk transitive . Moreover, each ̂ is dense of , since it intersects each . Thus, by using Baire Category theorem, we get, is a dense subset of . Clearly, the following Proposition is implied from combining Lemma 3.4 and Proposition 2.5. Then is -codisk-cyclic transitive, hence is a −codisk-cyclic operator.
Proof: Let and be two relatively open sets in . Since and are dense sets in , then from the condition (2), there are ∈ ⋂ and ∈ ⋂ , such that for some sequence { } in ℕ and 0 < ℰ < 1,

4-Conclusions:
A bounded linear operator on subspace, say , of an infinite dimensional complex separable Hilbert space ℋ is said to be -codisk-cyclic if satisfy: ∩ { : ∈ ℂ, | | ≥ 1, ∈ ℕ}, is dense in , for some nonzero vector in ℋ. This paper, presented two new concepts,codisk-cyclic and -codisk-cyclic transitive. We prove that their existence of a -codiskcyclic operator on -dimensional complex Hilbert space, also prove a criterion of -codiskcyclic operator in infinite dimensional. Finally, we discussed the relation between these two concepts.