67 Common diskcyclic vectors Nareen Bamerni Department of Mathematics, University of Duhok, Kurdistan Region, Iraq nareen.sabih@uod.ac Abstract In this paper, we study the common diskcyclic vectors for a path of diskcyclic operators. In particular, if {𝑇𝑑 : 𝑑 ∈ [π‘Ž, 𝑏]} is a path of diskcyclic operators, we show that under certain conditions the intersection of diskcyclic vectors for theses operators is a dense GΞ΄ set. Keywords: Diskcyclic operators, Common diskcyclic vectors, Weighted shift operators. 1. Introduction π‘‚π‘Ÿπ‘(𝑇, π‘₯) = {𝑇𝑛π‘₯: 𝑛 ∈ β„•} which is dense in 𝑋. The study of hypercyclic operators on a Banach space goes back to a 1969 paper of Rolewi Let 𝑋 be a Banach space and 𝐡(𝑋) be the space of all bounded linear operators on𝑋. An operator 𝑇 ∈ 𝐡(𝑋) is called hypercyclic if there is a vector π‘₯ ∈ 𝑋 called hypercyclic vector for 𝑇 such that cz [1] that proves if 𝐡 is the backward shift on the sequence space 𝑙𝑝(β„•) of then πœ†π΅ is hypercyclic whenever πœ† is a scalar of modulus > 1. Perhaps, inspired by Rolewicz example, Hilden and Wallen [2] considered the scaled orbit of an operator. An operator 𝑇 is supercyclic if there is a vector π‘₯ ∈ 𝑋 called supercyclic vector for 𝑇 such that β„‚π‘‚π‘Ÿπ‘(𝑇, π‘₯) = {πœ†π‘‡π‘› π‘₯: πœ† ∈ β„‚, 𝑛 ∈ β„•} is dense in 𝑋. Also, an operator 𝑇 is called diskcyclic if there is a vector π‘₯ ∈ 𝑋 called diskcyclic vector for 𝑇 such that the disk orbit π”»π‘‚π‘Ÿπ‘(𝑇, π‘₯) = {πœ†π‘‡π‘›π‘₯: πœ† ∈ β„‚, |πœ†| ≀ 1, 𝑛 ∈ β„•} is dense in 𝑋 [3]. For more information on these operators, the reader may refer to [4- 6]. Recently, the orbit of an operator in subspaces was studied. More precisely, if the orbit of an operator is dense in a subspace, then such an operator is called subspace-hypercyclic. By, the same manner if the scaled orbit (disk orbit) of an operator is dense in a subspace, then such an operator is called subspace-supercyclic (subspace-diskcyclic) respectively. For more information on these operators, the reader may refer to [7- 10]. Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/34.3.2679 Article history: Received 5 November 2020, Accepted 11 April 2021, Published in 2021. mailto:nareen.sabih@uod.ac Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(3)2021 68 of real numbers, then, the set {𝑇𝑑 : 𝑑 ∈ [π‘Ž, 𝑏]} is called a path of operators if 𝑇𝑑 ∈ 𝐡(𝑋) for It has been studied that under certain conditions, an uncountable family of hypercyclic operators (or supercyclic operators) has a dense GΞ΄ set of common hypercyclic vectors (or supercyclic vectors, respectively). For example, [11] ( [12] ) gave some conditions on a path of supercyclic ( hypercyclic ) operators to have a common supercyclic (or hypercyclic, respectively) vectors. For more information on common hypercyclic and supercyclic vectors, the reader may refer to [13- 19]. Now, since the set of all diskcyclic vectors is dense GΞ΄, then a countable collection of diskcyclic operators, by applying the Baire category theorem, has a dense GΞ΄ set of common diskcyclic vectors. However, it is unknown in which cases an uncountable family of diskcyclic operators has a common diskcyclic vectors. Therefore, in this paper, we study the common diskcyclic vectors for some uncountable families of diskcyclic operators. In particular, we give a sufficient condition for a path of operators to have a dense GΞ΄ set of common diskcyclic vectors, every operator in this path satisfies diskcyclic criterion. Then, we study a path of unilateral weighted backward shifts with common diskcyclic vectors. First, we recall the following definition from [12]. Definition 1.1. Let 𝑋 be a Banach space and [π‘Ž, 𝑏] be an interval all 𝑑 ∈ [π‘Ž, 𝑏] and if the map 𝑇: [π‘Ž, 𝑏] β†’ (𝐡(𝑋), β€–. β€– ) defined by 𝑇(𝑑) = 𝑇𝑑 is continuous with respect to both the operator norm topology on 𝐡(𝑋) and the usual topology on ℝ. 2. Main Results Definition 2.1. Let {𝛼𝐹𝑑 𝑛: 𝛼 ∈ 𝔻, 𝑛 β‰₯ 1} βŠ‚ 𝐡(𝑋) for each 𝑑 ∈ [π‘Ž, 𝑏]. Let 𝑑 β†’ 𝛼𝐹𝑑 𝑛 be a path of operators on [π‘Ž, 𝑏], then the set of common diskcyclic vectors for the path of operators is defined as follows: β‹‚ 𝐷𝐢(𝐹𝑑 ) = {π‘₯ ∈ 𝑋: π‘‚π‘Ÿπ‘(𝐹𝑑 , π‘₯), 𝑑 ∈ [π‘Ž, 𝑏]} π‘‘βˆˆ[π‘Ž,𝑏] is dense in 𝑋. Theorem 2.2. The set β‹‚ 𝐷𝐢(𝐹𝑑 )π‘‘βˆˆ[π‘Ž,𝑏] of common diskcyclic vectors is dense GΞ΄ set in 𝑋 if and only if for each nonempty open sets π‘ˆ and 𝑉, there exists a partition 𝑃 = {π‘Ž = 𝑑0 < 𝑑1 < β‹― < π‘‘π‘˜ = 𝑏} of [π‘Ž, 𝑏], 𝛼1, 𝛼2, … , π›Όπ‘˜ ∈ 𝔻, 𝑛1, 𝑛2, … , π‘›π‘˜ ∈ β„•, and an open set 𝐺 such that if 1 ≀ 𝑖 ≀ π‘˜ and 𝑑 ∈ [π‘‘π‘–βˆ’1, 𝑑𝑖 ] then 𝐺 βŠ† π‘ˆ and 𝛼𝑖 𝐹𝑑 (𝑛𝑖)𝐺 βŠ† 𝑉. Proof. The proof follows the same idea of [12, Theorem 2.1], therefore we omit the details. The following theorem shows that in some cases, an uncountable family of diskcyclic operators has a common diskcyclic vectors which is a dense GΞ΄ set. Theorem 2.3. Let 𝑋 be a separable, infinite dimensional Banach space, and let Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(3)2021 69 {𝐹𝑙 : 𝑙 ∈ [π‘Ž, 𝑏]} be a path of non-trivial bounded linear operators on 𝑋. If there exists a dense set 𝐷1 such that for every 𝑦 ∈ 𝐷1 and > 0, there exists 𝛿 > 0, a dense set 𝐷2, an increasing sequence of positive integers {π‘šπ‘— }𝑗=1 ∞ , and a set of maps {𝑆𝑙,𝑗 : 𝐷1 β†’ 𝑋: 𝑙 ∈ [π‘Ž, 𝑏], 𝑗 β‰₯ 1}such that 1. For each 𝑝 ∈ [π‘Ž, 𝑏] and π‘₯ ∈ 𝐷2, the sequence ‖𝐹𝑙 (π‘šπ‘—) π‘₯β€– ‖𝑆𝑝,𝑗 𝑦‖ β†’ 0 for all 𝑙 ∈ [π‘Ž, 𝑏] and 𝑗 β†’ ∞, 2. For each 𝑝 ∈ [π‘Ž, 𝑏], ‖𝑆𝑝,𝑗 𝑦‖ β†’ 0 as 𝑗 β†’ ∞, 3. For each 𝑝 ∈ [π‘Ž, 𝑏] and integer 𝑐 β‰₯ 1, there exists 𝑗 β‰₯ 𝑐 such that if |𝑙 βˆ’ 𝑝| < 𝛿 then ‖𝐹 𝑙 (π‘šπ‘—) 𝑆𝑝,𝑗 𝑦 βˆ’ 𝑦‖ < . Then β‹‚ DC(Ft)t∈[a,b] of common diskcyclic vectors is dense GΞ΄. Proof. Let U1and U2 be two non-trivial open sets in X. Pick y ∈ D1{0} and Οƒ > 0 such that B(y, Οƒ) βŠ† U2. Then there is an increasing sequence {mj}j=1 ∞ of positive integers, a dense set D2, Ξ΄ > 0 and a set of maps Sl,j: D1 β†’ X which satisfy the conditions (1), (2) and (3) with respect to the vector y and Ξ΅ = min { Οƒ 3 , β€–yβ€– 2 }. Let P = {a = l0 < l1 < β‹― < lk = b} be a partition of [a, b] where max{|li βˆ’ liβˆ’1|: 1 ≀ i ≀ n} < Ξ΄. Claim. Let 𝑖 be an integer such that 1 ≀ 𝑖 ≀ 𝑛, 𝐺 be a nontrivial open set, and 𝑐 β‰₯ 1 be an integer. Then there exists a nonempty open set 𝐺 β€² βŠ† 𝐺, a number 0 < πœ† ≀ 1, and an integer 𝑗 β‰₯ 𝑐 such that πœ†πΉ 𝑙 (π‘šπ‘—) 𝐺 β€² βŠ† π‘ˆ2, whenever 𝑙 ∈ [liβˆ’1, li]. Proof of Claim. Let 𝑀 ∈ 𝐷2 and π‘˜ be a small enough positive integer such that 𝐡(𝑀, π‘˜) βŠ† 𝐺.By putting 𝑝 = 𝑙𝑖 in conditions (1), (2) and (3), one can see that there exists 𝑗 β‰₯ 𝑐 such that ‖𝐹 𝑙 (π‘šπ‘—) 𝑀‖ ‖𝑆𝑙𝑖,𝑗 𝑦‖ < πœ€π‘˜ 2 for all 𝑙 ∈ [liβˆ’1, li], (1) ‖𝑆𝑙𝑖,𝑗 𝑦‖ < π‘˜ 2 , (2) and ‖𝐹 𝑙 (π‘šπ‘—) 𝑆𝑙𝑖,𝑗 𝑦 βˆ’ 𝑦‖ < for all 𝑙[liβˆ’1, li]. (3) Since Ξ΅ = min { Οƒ 3 , β€–yβ€– 2 } and 𝑦 β‰  0, then 𝐹 𝑙 (π‘šπ‘—) 𝑆𝑙𝑖,𝑗 𝑦 β‰  0 by (3). Now, let πœ† = 2 π‘˜ ‖𝑆𝑙𝑖,𝑗 𝑦‖, then it is clear that 0 < πœ† ≀ 1 by (2). Let = 𝑀 + 1 πœ† 𝑆𝑙𝑖,𝑗 𝑦 , 𝛼 = 𝑠𝑒𝑝 {πœ†πΉπ‘™ (π‘šπ‘—) : 𝑙 ∈ [liβˆ’1, li]} > 0 and 𝐺 β€² = 𝐡(𝑧, πœ€ 𝛼 ) ∩ 𝐺 βŠ† 𝐺. To prove that the open set 𝐺 β€² is nonempty, it is clear that ‖𝑧 βˆ’ 𝑀‖ = β€– 1 πœ† 𝑆𝑙𝑖,𝑗 𝑦 β€– = π‘˜ 2 < π‘˜, therefore 𝑧 ∈ 𝐡 (𝑧, πœ€ 𝛼 ) ∩ 𝐡(𝑀, π‘˜) βŠ† 𝐡 (𝑧, πœ€ 𝛼 ) ∩ 𝐺 = 𝐺 β€². Now, if 𝑙 ∈ [liβˆ’1, li], then β€–πœ†πΉπ‘™ (π‘šπ‘—) 𝑧 βˆ’ 𝑦‖ = β€–πœ†πΉ 𝑙 (π‘šπ‘—) 𝑀 + 𝐹 𝑙 (π‘šπ‘—) 𝑆𝑙𝑖,𝑗 𝑦 βˆ’ 𝑦‖ ≀ πœ† ‖𝐹 𝑙 (π‘šπ‘—) 𝑀‖ + ‖𝐹 𝑙 (π‘šπ‘—) 𝑆𝑙𝑖,𝑗 𝑦 βˆ’ 𝑦‖ < πœ† ‖𝐹 𝑙 (π‘šπ‘—) 𝑀‖ + by (3) = 2 π‘˜ ‖𝐹 𝑙 (π‘šπ‘—) 𝑀‖ ‖𝑆𝑙𝑖,𝑗 𝑦‖ + < 2 π‘˜ πœ€π‘˜ 2 + by (1) = 2 . And so if 𝑔 ∈ 𝐺 β€² and 𝑙 ∈ [liβˆ’1, li], then Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(3)2021 70 β€–πœ†πΉ 𝑙 (π‘šπ‘—) 𝑔 βˆ’ 𝑦‖ = β€–πœ†πΉ 𝑙 (π‘šπ‘—) 𝑔 βˆ’ πœ†πΉ 𝑙 (π‘šπ‘—) 𝑧 + πœ†πΉ 𝑙 (π‘šπ‘—) 𝑧 βˆ’ 𝑦‖ ≀ πœ† β€– 𝐹 𝑙 (π‘šπ‘—) β€– ‖𝑔 βˆ’ 𝑧‖ + β€–πœ†πΉ 𝑙 (π‘šπ‘—) 𝑧 βˆ’ 𝑦‖ < 𝑠𝑒𝑝 {πœ† β€– 𝐹 𝑙 (π‘šπ‘—) β€– : 𝑙 ∈ [liβˆ’1, li]} 𝛼 + 2 < 3 < 𝜎. Therefore, πœ†πΉ 𝑙 (π‘šπ‘—) (𝐺 β€²) βŠ† 𝐡(𝑦, 𝜎) βŠ† π‘ˆ2, whenever 𝑙 ∈ [liβˆ’1, li]. The claim is proved. Returning to the proof of the theorem, Let 𝐺0 be an open ball with center 𝑏, 𝑏 ∈ 𝐷2 such that 𝐺0 βŠ† π‘ˆ1. Then by claim, there are a scalar πœ†1such that 0 < πœ†1 ≀ 1, a nonempty open set 𝐺1 βŠ† 𝐺0, and a positive integer 𝑗1 β‰₯ 1 such that πœ†1𝐹𝑙 (π‘šπ‘—1 )(𝐺1) βŠ† π‘ˆ2 whenever 𝑙 ∈ [l0, l1]. Again, by the claim, there is a scalar πœ†2such that 0 < πœ†2 ≀ 1, a nonempty open set 𝐺2 βŠ† 𝐺1, and a positive integer 𝑗2 β‰₯ 𝑗1 such that πœ†2𝐹𝑙 (π‘šπ‘—2 )(𝐺2) βŠ† π‘ˆ2 whenever 𝑙 ∈ [l1, l2]. By applying this process 𝑛 times, we get open sets 𝐺𝑛 βŠ† πΊπ‘›βˆ’1 βŠ† β‹― βŠ† 𝐺1 βŠ† 𝐺0, scalars 0 < πœ†1, πœ†2, … , πœ†π‘› ≀ 1, and integers 1 ≀ 𝑗1 < 𝑗2 < β‹― < 𝑗𝑛 such that πœ†π‘– 𝐹𝑙 (π‘šπ‘—π‘– ) (𝐺𝑖 ) βŠ† π‘ˆ2 whenever 1 ≀ 𝑖 ≀ 𝑛 and 𝑙 ∈ [liβˆ’1, li]. So, 𝐺𝑛 βŠ† π‘ˆ1 and πœ†π‘– 𝐹𝑙 (π‘šπ‘—π‘– ) (𝐺𝑛) βŠ† πœ†π‘– 𝐹𝑙 (π‘šπ‘—π‘– ) (𝐺𝑖 ) βŠ† π‘ˆ2 whenever 𝑙 ∈ [liβˆ’1, li]. The proof follows from Theorem 2.2. Even though not every diskcyclic operator satisfies diskcyclic criterion, the following proposition shows that in some cases every diskcyclic operator satisfies diskcyclic criterion. Proposition 2.4. If a path {𝐹𝑑 : 𝑙 ∈ [π‘Ž, 𝑏]} satisfies the conditions of Theorem 2.3., then every that {πœ†π‘– ∈ 𝔻\{0}: 𝑖 β‰₯ 1} be a countable set. Suppose that the sequence {𝑇𝑛} is an enumeration of the set {πœ†π‘– 𝐹𝑙 𝑗 : 𝑖, 𝑗 β‰₯ 1} for each 𝑙 ∈ [π‘Ž, 𝑏].Now, suppose that {𝑇𝑛} satisfies the operator 𝐹𝑙 satisfies the diskcyclicity criterion. Proof. Suppose universality criterion, and then each 𝐹𝑙 satisfies the diskcyclic criterion; see [20, Definition 1.1], [5, Theorem 1.2, Theorem 2.6, Proposition 2.8]. Thus, it is enough to prove that {𝑇𝑛} satisfies the universality criterion which is equivalent to showing that for every nonempty open sets π‘ˆ, 𝐺, π‘Š with 0 ∈ π‘Š, we have 𝑇𝑛(π‘ˆ) ∩ π‘Š β‰  βˆ… and 𝑇𝑛(π‘Š) ∩ 𝐺 β‰  βˆ…; see [20, Theorem 3.4]. Now, by Theorem 2.3, for any dense set 𝐷1, we can choose 𝑦 ∈ 𝐷1{0} and 0 < < ‖𝑦‖ such that 𝐡(𝑦, ) βŠ† 𝐺 π‘Žπ‘›π‘‘ 𝐡(0, ) βŠ† π‘Š. Then there exists an increasing sequence {mj}j=1 ∞ of positive integers, a dense set 𝐷2, a small positive number 𝛿 > 0, and maps {𝑆𝑐,𝑗 : 𝐷1 β†’ 𝑋: 𝑗 β‰₯ 1, 𝑐 ∈ [π‘Ž, 𝑏]} satisfying all conditions of Theorem 2.3. Thus, we can choose π‘₯ ∈ 𝐷2 ∩ π‘ˆ, an integer 𝑗 β‰₯ 1 such that ‖𝐹 𝑙 (π‘šπ‘—) π‘₯β€– ‖𝑆𝑙,𝑗 𝑦‖ < πœ€2 2 , ‖𝐹 𝑙 (π‘šπ‘—) 𝑆𝑙,𝑗 𝑦 βˆ’ 𝑦‖ < and ‖𝑆𝑙,𝑗 𝑦‖ β†’ 0 Let πœ† ∈ {πœ†π‘– : 𝑖 β‰₯ 1}, since πœ† < 1 we can assume that πœ† < 2‖𝑆𝑙,𝑗 𝑦‖ < 2 πœ† (4) Then πœ†πΉ 𝑙 (π‘šπ‘—) = 𝑇𝑛 for some 𝑛 β‰₯ 1. Since π‘₯ ∈ π‘ˆ, then by Equation (4), we get Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(3)2021 71 ‖𝑇𝑛π‘₯β€– = β€–πœ†πΉπ‘™ (π‘šπ‘—) π‘₯β€– < 2 πœ€ ‖𝐹 𝑙 (π‘šπ‘—) π‘₯β€– ‖𝑆𝑙,𝑗 𝑦‖ < 2 πœ€ πœ€2 2 = Then, 𝑇𝑛 (π‘ˆ) ∩ π‘Š β‰  βˆ…. Again from Equation (4), we have 1 πœ† ‖𝑆𝑙,𝑗 𝑦‖ < which means that 𝑆𝑙,𝑗 𝑦 ∈ π‘Š. Also, we have ‖𝑇𝑛(𝑆𝑙,𝑗 𝑦) βˆ’ 𝑦‖ = ‖𝐹𝑙 (π‘šπ‘—) 𝑆𝑙,𝑗 𝑦 βˆ’ 𝑦‖ < So, 𝑇𝑛(π‘Š) ∩ 𝐺 β‰  βˆ…, which gives the proof. Corollary 2.5. An operator 𝑇 ∈ 𝐡(𝑋) satisfies diskcyclic criterion if and only if there exists a dense set 𝐷1 such that for each 𝑦 ∈ 𝐷1and a small positive number > 0, there is an increasing sequence {mj}j=1 ∞ of positive integers, a dense set 𝐷2 and maps 𝑆𝑗 : 𝐷1 β†’ 𝑋 satisfying 1. For each π‘₯ ∈ 𝐷2, the sequence β€– 𝑇 π‘šπ‘— π‘₯‖‖𝑆𝑗 𝑦‖ β†’ 0 as 𝑗 β†’ ∞, 2. ‖𝑆𝑗 𝑦‖ β†’ 0 as 𝑗 β†’ ∞, 3. For each integer 𝑐 β‰₯ 1, there exists 𝑗 β‰₯ 𝑐 such that β€–π‘‡π‘šπ‘— 𝑆𝑗 𝑦 βˆ’ 𝑦‖ < . Since a unilateral shift 𝑇 is diskcyclic if and only if it is hypercyclic with 𝐻𝐢(𝑇) = 𝐷𝐢(𝑇) [5, Corollary 3.6], then the following propositions follow immediately by [12, Theorem 3.5] and [12, Theorem 4.1] respectively. Proposition 2.6. Suppose that 𝑇 and 𝑆 are two diskcyclic weighted backward shifts. Then , there exists a path of unilateral weighted backward shifts between 𝑇 and 𝑆 , such a path has a dense GΞ΄ set of common diskcyclic vectors. Proposition 2.7. Suppose that 𝑇 and 𝑆 are two diskcyclic weighted backward shifts. Then , there exists a path of unilateral weighted backward shifts between 𝑇 and 𝑆 , such a path has no any common diskcyclic vector. Conclusion We have given a sufficient condition for a path of operators to have a dense GΞ΄ set of common diskcyclic vectors such that every operator in that path satisfies diskcyclic criterion. References 1. Rolewicz, S. On orbits of elements. Studia Mathematica, 1969 ,1(32):17-22,. 2. Hilden, H. ; Wallen, L. Some cyclic and non-cyclic vectors of certain operators. Indiana University Mathematics Journal, 1974, 23(7):557-565. 3. Zeana, J. Cyclic Phenomena of operators on Hilbert space. PhD thesis, Thesis, University of Baghdad, 2002. 4. Bamerni, N. ; Kilicman, A. Operators with diskcyclic vectors subspaces. Journal of Taibah University for Science, 2015, 9(3):414-419, 5. Bamerni, N. ; Kilicman, A. ; Noorani, M. S. M. A review of some works in the theory of diskcyclic operators. Bulletin of the Malaysian Mathematical Sciences Society, 2015, 1-17. 6., Bayart F. ; Matheron, E. Dynamics of linear operators, volume. Cambridge university press, 2009, 179. 7. Bamerni, N. ; Kadets, V. ; Kilicman, A. Hypercyclic operators are subspace hypercyclic. Journal of Mathematical Analysis and Applications, 2016, 435,2,1812-1815. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(3)2021 72 8. Bamerni, N. ; Kilicman, A. On subspaces diskcyclicity, Arab Journal of Mathematical Sciences, 2015, 23(2),133-140,. 9. Madore, B. F. ; Martinez-Avendano, R. A. Subspace hypercyclicity. Journal of Mathematical Analysis and Applications, 2011, 373(2):502-511,. 10. Xian-Feng, Z.; Yong-Lu, S. ; Yun-Hua. Z. Subspace-supercyclicity and common subspace supercyclic vectors., Journal of East China Normal University, 2012, 1(1):106- 112. 11. Chan, K. C. ; Sanders, R. Common supercyclic vectors for a path of operators. Journal of Mathematical Analysis and Applications, 2008, 337(1):646-658. 12. Chan, K. C. ; Sanders, R. Two criteria for a path of operators to have common hypercyclic vectors. Journal of Operator Theory, 2009, 61(1):191-223. 13. Abakumov, E. ; Gordon, J. Common hypercyclic vectors for multiples of backward shift. Journal of Functional Analysis, 2003, 200(2):494-504,. 14. Aron, R.; Bes, J.; Leon, F. ; Peris, A. Operators with common hypercyclic subspaces. Journal of Operator Theory, 2005,54(2):251-260,. 15. Bayart, F. Common hypercyclic subspaces. Integral Equations and Operator Theory, 2005, 53(4):467-476,. 16. Bayart, F. ; Matheron, E. How to get common universal vectors. Indiana University mathematics journal, 2007, 56(2):553-580. 17. Sanders, R. Common hypercyclic vectors and the hypercyclicity criterion. Integral Equations and Operator Theory, 2009, 65(1):131-149. 18. Shkarin, S. Remarks on common hypercyclic vectors. Journal of Functional Analysis, 2010, 258(1):132-160. 19. Zhang, L. ; Zhou, Z.-H. Notes about the structure of common supercyclic vectors. Journal of Mathematical Analysis and Applications, 2014, 418(1):336-343. 20. Bernal-Gonzalez, L. ; Grosse-Erdmann, K.-G. The hypercyclicity criterion for sequences of operators. Studia Mathematica, 2003, 157:1.