Disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type operators

We characterize disjointness of supercyclic operators which map a holomorphic function to a partial sum of the Taylor expansion. In particular, we show that disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type operators. Moreover, we give a su cient condition to yield the disjoint supercyclicity for families of Taylor-type operators.


Introduction
Let X, Y be two topological vector space over R or C. A sequence of linear and continuous operators T n ∶ X → Y, n = , , . . . is said to be hypercyclic if there exists a vector x ∈ X such that {T x, T x, . . .} is dense in Y . Such a vector x is called a hypercyclic vector for {T n } n∈N . If the sequence {T n } n∈N comes from the iterates of a single operator T ∶ X → Y, i.e. T n = T n , n = , , . . . , then T is called hypercyclic.
In 1974, Hilden and Wallen introduced in [1] the notion of supercyclicity. They showed that all unilateral weighted backward shifts are supercyclic, but no vector is supercyclic for all unilateral weighted backward shifts. Recall that a sequence of linear and continuous operators T n ∶ X → Y, n = , , . . . is said to be supercyclic provided there exists a vector x ∈ X such that {αT x, αT x, . . . ∶ α ∈ C} is dense in Y . Such a vector x is called a supercyclic vector for {T n } n∈N . Good sources of background information on hypercyclic and supercyclic operators include [2][3][4].
Obviously, by the de nition, the following diagram holds true in the disjoint setting: Disjoint hypercyclicity ⇒ Disjoint supercyclicity.
First, we introduce some standard notations and terminology. The set of holomorphic functions on a simply connected domain Ω ⊂ C, to be denoted H(Ω), becomes a complete topological vector space under the topology inherited by the uniform convergence on all the compact subsets of Ω. Moreover, for any compact set K ⊂ C, we denote A(K) = {g ∈ H(K ) ∶ g is continuous on K}, For a function g de ned on K, we use the notation g K = sup z∈K g(z) . Now for every K ∈ M Ω and every sequence of natural numbers {λ n } n∈N we consider the sequence of operators: (z−ζ ) k denote the nth partial sum of the Taylor series of f with center ζ . f is said to belong to the collection U(Ω, ζ ) of functions with universal Taylor series expansions around ζ whenever {T [12,13] had shown that the collection U(Ω, ζ ) is a dense G δ subset of H(Ω), and U(Ω, ζ ) ≠ ∅ for any simply connected domain Ω and any point ζ ∈ Ω. Indeed, he proved that if the sequence {λ n } n∈N is unbounded then the corresponding sequence of operators {T (ζ ) λ n } n∈N is hypercyclic. Costakis and Tsirivas [14] provided a new strong notion of universality for Taylor series called doubly universal Taylor series. Chatzigiannakidou and Vlachou [15] dealt with the existence of doubly universal Taylor series de ned on simply connected domains with respect to any center, which generalized the results of Costakis and Tsirivas for the unit disk. Moreover, Chatzigiannakidou [16] studied some approximation properties of doubly universal Taylor series de ned on a simply connected domain Ω.
In order to research the disjointness of hypercyclicy for families of Taylor-type operators directly, Vlachou [17] introduced a class.
As we all know, the functions of the above class are disjoint hypercyclic vectors, so if we want to research some characterizations of disjointness of hypercyclicity, we consider this class as empty or non-empty. It is clear that the sequences of natural numbers {λ (σ) n } n∈N play a key role in the study of this class. In this paper, we require a special de nition of {λ    [17] showed that there exists a rearrangement {λ (π(σ)) n } n∈N , which is well ordered. Thus, in this paper we assume that we have a well ordered nite collection of sequences of natural numbers {λ (σ) n } n∈N .
Following the same path as [15], Vlachou [17] showed a necessary and su cient condition for families of taylor-type operators to be disjoint hypercyclic as follows: Inspired by [17], we introduce another class to research the disjointness of supercyclicity for operators which map a holomorphic function to a partial sum of the Taylor expansion.
n } n∈N be a nite collection of sequences of natural numbers. If for every choice of compact sets K , K , . . . , K σ ∈ M Ω the set The paper is organized in the following manner: In section 2, we obtain that Disjoint hypercyclicity ⇐⇒ Disjoint supercyclicity for families of taylor-type operators. In section 3, we provide a su cient condition to get the disjointness of supercyclic operators who map a holomorphic function to a partial sum of the Taylor expansion.

disjoint hypercyclicity equals disjoint supercyclicity
In this section, we prove that Disjoint hypercyclicity ⇐⇒ Disjoint supercyclicity for families of taylortype operators. In order to prove the main theorem, we need some fundamental knowledge about thinness.

De nition 2.1 ([18, Chapter 5])
. Let S be a subset of C and ξ ∈ C. Then S is non-thin at ξ if ξ ∈ S {ξ} and if for every subharmonic function u de ned on a neighbourhood of ξ,

Otherwise we say that S is thin at ξ.
Thinness is obviously a local property, i.e. S is non-thin at ξ if and only if U ∩ S is non-thin at ξ for each open neighbourhood U of ξ. If two sets are both thin at a particular point, so is their union.

Lemma 2.2 ([17, Lemma 2.2]).
Let Ω ⊂ C be a simply connected domain. Then there exists an increasing sequence of compact sets E k , k = , , . . . with the following properties:  (ii)⇒(iii): Choose an increasing sequence of compact sets E k as stated in Lemma 2.2. Since V we may also x a strictly increasing sequence of natural Clearly lim n must have nite terms and which implies that for some constant C > , }. At least one of the above sets is in nite. Without loss of generality, we assume that I is in nite. Let p k (z) be de ned by Obviously, Then E is closed and non-thin at ∞. Let z ∈ E. Then for large enough k, z ∈ E k and z − ζ ≥ R. By (1) and (2), since I is in nite and k large enough, we obtain k ∈ I and thus p k → compactly on C. Let ξ ∈ ∂Ω with ξ − ζ = R, then from the above which contradicts (3). Now we prove the case J is in nite, we set Then following the same argument as I is in nite, we arrive at contradiction.

A su cient condition for disjoint supercyclicity
In this section, we present a su cient condition to imply the disjointness of supercyclicity for Taylor-type operators, which is di erent from Theorem 2.3. Any continuous function f can be approximated uniformly on a compact subset K of C by polynomials provided that C∖K is connected and f extends to be holomorphic on a neighbourhood of K. Ransford [18] gave a somewhat stonger version of this result called Bernstin-walsh Theorem. Vlachou [17] generalized Bernstinwalsh Theorem. On this basis, we give minor modi cations. Though the proof is similar to the above two papers, for the convenience of the reader we give the details of the proof. Write deg(p) as the degree of a polynomial p. Proof. The proof is divided into two cases. Case : c(K) > . Γ is a closed contour in U ∖ K which winds once around each point of K and zero times round each point of C ∖ U. Since lim n→∞ τ n = +∞, we can choose n large enough to ensure τ n ≥ . Thus we can consider a Fekete polynomial q τ n of degree τ n for K, for ω ∈ K de ne

De nition 3.1. Let h n
Obviously, deg(p n ) ≤ τ n − . Cauchy's integral formula gives and hence, where l(Γ ) is the length of Γ and dist(Γ , K) is the distance of Γ from K. Since f n is {σ n }−locally bounded, there exists a positive constant A > such that f n Γ ≤ A σ n . In addition, according to the proof of Theorem 6.3.1 in [18], we see that Case : c(K) = . Let (K k ) k≥ be a decreasing sequence of non-polar compact subsets of U, with connected complements, such that lim k→∞ K k = K. Let θ k denote the corresponding numbers de ned in the theorem, as shown in case 1 lim sup n d τ n (f n , K) τn ≤ lim sup n d τ n (f n , K k ) τn ≤ Cθ k . Now we prove that lim k→∞ θ k = . De ne the function Thus (h k ) k≥ is an increasing sequence of harmonic functions on z ∈ C ∖ K and h k (∞) → ∞, Harnack's Theorem implies that h k → ∞ locally uniformly on C ∖ K . In particular g C ∞ ∖K k (z, ∞) → ∞ uniformly on C ∞ ∖ U, which shows that lim k→∞ θ k = .
For convenience, we de ne γ = Cθ in the following. Proof. Suppose {f j } j∈N is an enumeration of polynomials with rational coe cients. In view of [13], there exists a sequence of compact sets {K m } m∈N in M Ω , such that for every K ∈ M Ω is contained in some K m . For α ∈ C and every choice of positive integers s, n, m σ and j σ , let An application of Mergelyan's Theorem shows that Therefore, by Baire's Category Theorem, it is su cient to prove that Choose g ∈ H(Ω), ε > and a compact subset L of Ω. Without loss of generality, we may assume that L has connected complement, ζ ∈ L . For every β < , Runge's Theorem implies that we may x a polynomial p such that: Moreover, we x two open and disjoint sets U , U with L ⊂ U and ∪ σ σ= Next, our proof is divided into two steps: Step . For σ ≥ , we will construct a sequence of polynomials {Q (σ) n } n∈N via a nite induction with the following properties: We de ne a function f n as Moreover, let σ n = λ (σ− ) n and τ n = λ (σ) n and a compact setK ⊂ U − ζ . First of all, we prove σ = case. Note that where c = f j σ − βp K +ζ . So {f n } n∈N is {σ n }−locally bounded. By Proposition 3.2, it follows that there exists γ > such that lim sup n d τ n (f n , K) τn ≤ γ.
This implies that we can x a sequence of polynomials p n with degree less or equal to τ n so that for n su ciently large.
Thirdly, we will show σ(σ ≥ ) case. Since βQ , it follows that for n large enough, Hence, by the de nition of f n , we see that where c = f j σ − p K +ζ . Similarly to σ = , we can x a sequence of polynomials p n with degree less or equal to τ n such that: We set Q n } n∈N via a nite induction with the above three properties.