On Hereditarily Codiskcyclic Operators

: Many codiskcyclic operators on infinite-dimensional separable Hilbert space do not satisfy the criterion of codiskcyclic operators. In this paper, a kind of codiskcyclic operators satisfying the criterion has been characterized, the equivalence between them has been discussed and the class of codiskcyclic operators satisfying their direct summand is codiskcyclic. Finally, this kind of operators is used to prove that every codiskcyclic operator satisfies the criterion if the general kernel is dense in the space.


Introduction:
Let be an infinitedimensional separable Hilbert space, the unit disk is denoted by , is the unit ball, and ( ) is the set of all linear bounded operators onto . Hilden and Wallen in 1974 introduced the concept of supercyclic operator ∈ ( ), as there exists a non-zero vector in such that ( , ) ≔ { : n ≥ 0, ∈ ℂ} is dense in 1 . After that in 2002, Jamil in her thesis divided ℂ into three areas according to the unit circle:  The interior of the unit circle: thus, an operator is called diskcyclic, if ( , ) ≔ { : n ≥ 0, ∈ } is dense in .  The unit circle: then the operator is called circle cyclic, if ( , ) ≔ { : n ≥ 0, | | = 1} is dense in .  The exterior of the unit circle: hence the operator is called codiskcyclic, if ( , ) ≔ { : n ≥ 0, ∈ } is dense in . She studied some of their properties like the range of them, some necessarily and sufficient conditions to be 2 . In 2004, Leon-Saavedra and Muller, proved that every circle cyclic operator is hypercyclic, which mean ( , ) ≔ { : n ≥ 0} is dense in 3 , while the other kinds have been gaining importance in recent years such as Liang and Zhou 4,5 , Wong and Zeng 6 …etc.
In 2002, Jamil 2 , introduced a criterion for codiskcyclic operators. She showed that there are codiskcyclic operators that do not satisfy this criterion. Thus, the natural question arises is which kind of codiskcyclic operators can satisfy the criterion?
This paper offered a partial solution to this problem by presenting a new kind of codiskcyclic operator that is called hereditarily codiskcyclic. By using the concept of hereditarily codiskcyclic, codiskcyclic operators have been proved to satisfy the criterion whenever the generalize kernel is dense in the space. { : ≥ 1, ∈ } = ( ) = In addition to discussing how big the set of codiskcyclic vectors is for { }, the following proposition studying the relation between { } is codiskcyclic and topologically transitive. Since the proof of 3) ⇒ 1) is trivial, thus it is omitted.

Proposition (2):
Let be a separable infinite dimensional complex Hilbert space and ∈ ( ), Let { } be a nonnegative integer sequence. Then the following statements are equivalent 1.

Hereditarily Codiskcyclic Operators:
It is well known that not every codiskcyclic operators satisfies codiskcyclic criterion, so this section introduces the following concept and argue the relation between these operators with codiskcyclic criterion.

Definition (2):
A bounded linear operator is called here ditarilty codiskcyclic if there is a sequence { } such that for all subsequence { } of { }, { } is codiskcyclic. Every hereditarily codiskcyclic is a codiskcyclic operator. One question raises is the converse true. The following definition and proposition are needed to answer this question. Thus, for suitable ∈ ℕ, there are = ∈ (1, ∞) and = ∈ ℕ such that ∩ ≠ ∅. Therefore, by proposition (2) is a codiskcyclic operator. The result now discusses the relation between hereditarily codiskcyclic operator and operator which satisfies codiskcyclic criterion. Jamil in 2 proved that if ⨁ =1 ∈ (⨁ =1 ) is a codiskcyclic, then is a codiskcyclic operater for all .

Remark (2):
By the same argument one can prove that part (2) of proposition (4) is true for = ∞. One of the applications of proposition (4) is to prove that codiskcyclic operator is hereditarily codiskcyclic under certain condition. But first the following known proposition is important to correctly interpret the results. For all ∈ , there are ∈ , ∈ ℕ such that = . Hence as → ∞ a. Therefore, by proposition (4), is a hereditarily codiskcyclic operator.

Conclusions:
The open problem is "Which kind of codiskcyclic operators satisfy codiskcyclic criterion? This paper introduced a hereditarily codiskcyclic operators, studied some characterization, and proved that every codiskcyclic operator satisfies the codiskcyclic criterion if the space contains dense general kernel set.
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