Simultaneous universality

arXiv:1701.07311v1 [math.FA] 25 Jan 2017 SIMULTANEOUS UNIVERSALITY L. BERNAL-GONZ´ ALEZ AND A. JUNG Abstract. In this paper, the notion of simultaneous universality is introduced, concerning operators having orbits that simultaneously approximate any given vector. This notion is related to the well known concepts of universality and disjoint universality. Several criteria are provided, and several applications to specific operators or sequences of operators are performed, mainly in the setting of sequence spaces or spaces of holomorphic functions. 1. Introduction In this paper, we are concerned with the phenomenon of simultaneous approximation by the action of several operators or, more generally, by the action of several sequences of mappings. When the existence of a dense orbit under an operator is proved, we are speaking about universality or hypercyclicity, see below. In many situations, it is possible to show the existence of one vector whose orbits under two or more operators approximate any given vector. Pushing the question quite further, we wonder under what conditions such approximation takes place by using a common subsequence. This, together with its connection with other kinds of joint universality, will make up the main aim of the present manuscript. Next, we fix some related notation and terminology to be used in this work. For a good account of concepts, results and history concerning hypercyclicity, the reader is referred to the books [2,21]. By N,N0,R,C,D, B(a, r),B(a, r) (a∈C, r > 0) we denote, respectively, the set of positive integers, the set N∪ {0}, the real line, the complex plane, the open unit disk {z∈C:|z|<1}, the open disk with center aand radius r, and the corresponding closed disk. Let X, Y be two Hausdorff topological spaces, and Tn:X→Y(n= 1,2,...) be a sequence of continuous mappings. Recall that (Tn) is said to be universal whenever there is some (Tn)-orbit which is dense in Y, that is, there exists an element x0∈X–called universal for (Tn)– such that {Tnx0:n∈N}=Y. Note that Ymust be separable. We denote by U((Tn)) the set of universal elements for (Tn). When X=Yand T:X→Xis a continuous self-mapping, 2010 Mathematics Subject Classification. 30E10, 47B33, 47A16, 47B38. Key words and phrases. hypercyclic operator, composition operator, disjoint universality, simultaneous universality. 1 2 BERNAL AND JUNG then Tis called universal provided that the sequence (Tn) of iterates of T(i.e., T1=T,T2=T◦T,T3=T◦T2, and so on) is universal, in which case the set U((Tn)) of universal elements will be denoted by U(T). A sequence Tn:X→Y (n= 1,2,...) of continuous mappings is said to be densely universal if U((Tn)) is dense in X. Birkhoff’s transitivity theorem asserts that, if Xis a Baire space (in particular, if Xis completely metrizable) and Yis second-countable (in particular, if Xis metrizable and separable), then (Tn) is densely universal if and only if (Tn) is transitive (that is, given nonempty open sets U⊂X,V⊂Y, there is N∈N with TN(U)∩V6=∅); if this is the case, then U((Tn)) is residual (in fact, a dense Gδsubset) in X. If Xlacks isolated points and T:X→Xis universal, then U(T) is dense in X(so residual if Xis, in addition, completely metrizable). In the case in which Xand Yare topological vector spaces over K(= Ror C) and (Tn)⊂L(X, Y ) := {linear continuous mappings X→Y}, the words hypercyclic and universal are synonymous, although hypercyclic is mostly used, as well as the alternative notation HC((Tn)) := U((Tn)) (and HC(T) := U(T) for T∈L(X) := L(X, X) = {operators on X}). In particular, we have if Xand Y are F-spaces with Yseparable, then HC((Tn)) (HC(T), with Xseparable, resp.) is residual in Xas soon as (Tn) is transitive (as soon as Tis hypercyclic, resp.). Recall that an F-space is a completely metrizable topological vector space. Assume now that X, Y are topological spaces, with Xa Baire space and Y second-countable, and that Sn:X→Yand Tn:X→Y(n∈N) are densely universal sequences. Since U((Sn)),U((Tn)) are dense Gδsubsets of X, we have that U((Sn)) ∩ U((Tn)) is also dense, so non-empty. Hence there is a common hypercyclic element x∈X. So, for a given point y∈Y, there are sequences {n1< n2<· · ·} and {m1< m2<· · · } in Nsuch that Snjx→yand Tmjx→yas j→ ∞. Then the following question arises naturally: Under what conditions on (Sn)and (Tn)one can guarantee the existence of an element x∈Xsuch that, for any given y∈Y, there is one sequence {n1< n2<· · · } ⊂ Nsuch that Snjx−→ y←− Tnjxas j→ ∞? Of course, a similar question can be posed for finitely many sequences and for finitely many single operators on X, just by considering the sequences of their iterates in the latter case. With this in mind, the new concept of simultaneous universality will be introduced in the next section, and compared to other related notions existing in the literature, such as those of disjoint hypercyclicity and the weakly mixing property. Several sufficient conditions for simultaneous universality/hypercyclicity will be provided in Section 3. Examples of finite families of simultaneous hypercyclic operators will be furnished in sections 4–6, starting with SIMULTANEOUS UNIVERSALITY 3 multiples of an operator and ending up in the frameworks of sequence spaces and of spaces of analytic functions on complex domains. 2. Simultaneously universal sequences Let us define the new concept that is the matter of this paper. If p∈Nand Yis a nonempty set, then by ∆(Yp) we denote the diagonal of Yp=Y× · · · × Y (ptimes), that is, the subset ∆(Yp) = {(y,y,...,y) : y∈Y}. If Yis a topological space, then Ypis assumed to be endowed with the product topology. Definition 2.1. Let p∈Nand X, Y be Hausdorff topological spaces. Assume that, for each j∈ {1,...,p},Tj,n :X→Y(n∈N) is a sequence of continuous mappings. Consider the sequence [T1,n,...,Tp,n] : x∈X7−→ (T1,nx,...,Tp,nx)∈Yp(n∈N). Let also T1,...,Tp:X→Xbe continuous mappings. (a) We say that the sequences (T1,n),...,(Tp,n) are simultaneously universal (or s-universal ) whenever there exists an element x0∈X–called s-universal for (T1,n),...,(Tp,n)– satisfying {[T1,n,...,Tp,n]x0:n∈N} ⊃ ∆(Yp). The set of such s-universal elements will be denoted by s-U((T1,n),...,(Tp,n)). (b) The sequences (T1,n),...,(Tp,n) are said to be densely simultaneously universal if the set s-U((T1,n),...,(Tp,n)) is dense in X. And they are called hereditarily simultaneously universal (hereditarily densely simultaneously universal, resp.) if, for every strictly increasing sequence (nk)⊂N, the sequences (T1,nk), . . ., (Tp,nk) are s-universal (densely s-universal, resp.). (c) The mappings T1, . . . , Tpare called s-universal (densely s-universal,hereditarily s-universal,hereditarily densely s-universal, resp.) if the sequences (Tn 1),...,(Tn p) are s-universal (densely s-universal, hereditarily s-universal, hereditarily densely s-universal, resp.). The set s-U((Tn 1),...,(Tn p)) of corresponding s-universal elements will be denoted by s-U(T1,...,Tp). Remarks 2.2. 1. If Yis first-countable (in particular, if Yis metrizable), then the s-simultaneous universality of (Tj,n)n∈N(1 ≤j≤p) means the existence of some x0∈Xenjoying the property that, for every y∈Y, there is a (strictly increasing) sequence (nk)⊂Nsuch that Tj,nkx0→yas k→ ∞ (j= 1,...,p). 2. In [18, Kapitel 1] the notion of relative universality on a closed subset of the arrival space is introduced under very general assumptions. In the present paper we study a special case of this situation (note that ∆(Yp) is closed in Ypsince Yp is Hausdorff) under more specific hypotheses. 3. According to the introduction, if X, Y are topological vector spaces and Tj,n, Tj∈ L(X, Y ) (j= 1,...,p;n∈N), then we use the expressions “s-hypercyclic”, 4 BERNAL AND JUNG “densely s-hypercyclic” and “hereditarily densely s-hypercyclic” rather than “suniversal”, “densely s-universal” and “hereditarily densely s-universal”, respectively. In addition, we will denote s-HC((T1,n),...,(Tp,n)) := s-U((T1,n),...,(Tp,n)) and s-HC(T1,...,Tp) := s-U(T1,...,Tp) in this case. 4. For a single operator T, hypercyclicity (hereditary hypercyclicity, resp.) is equivalent to dense hypercyclicity (hereditary dense hypercyclicity, resp.). 5. The property of simultaneous universality of (T1,n),...,(Tp,n) is weaker than the property that the sequence ([T1,n,...,Tp,n]) is subspace-universal for ∆(Yp), meaning that the set {[T1,n,...,Tp,n]x0:n∈N} ∩ ∆(Yp) is dense in ∆(Yp) for some x0∈X(see e.g. [1,22,24] for results on subspace-hypercyclicity/universality). Before going on, we want to compare s-universality to other related concepts defined in the literature. In 2007, B`es, Peris and the first author ([11],[4]) introduced the notion of disjoint (or d-) universality (sometimes called d-hypercyclicity in the mentioned references). Under the same assumptions and terminology as in Definition 2.1, the sequences (T1,n),...,(Tp,n) are said to be d-universal whenever the sequence [T1,n . . . , Tp,n] : X→Yp(n∈N) is universal, that is, whenever there exists some x0∈Xsuch that the joint orbit {(T1,nx0,...,Tp,nx0) : n∈N}is dense in Yp. As a matter of fact, d-universality should not be confused with the universality of the sequence T1,n ⊕ · · · ⊕ Tp,n : (x1,...,xp)∈Xp7−→ (T1,nx1,...,Tp,nxp)∈Yp. Trivially, disjoint universality of (T1,n),...,(Tp,n) implies universality of the last sequence as well as simultaneous universality of (T1,n),...,(Tp,n). Also, trivially, s-universality implies the universality of each sequence (Tj,n)n∈N(j= 1,...,p) (in particular, Ymust be separable). But no other implications among these properties hold, even considering only p= 2 and sequences of iterates of single operators. The following examples illustrate this situation: 1. Assume that Tis a hypercyclic operator on a topological vector space. Then the operators T, T are s-hypercyclic but not d-hypercyclic. 2. In 1969, S. Rolewicz [26] proved that if c∈Khas modulus >1 and B is the backward shift (xn)∈ℓ27→ (xn+1)∈ℓ2, then the operator cB is hypercyclic. In particular, the operators T= 2Band S= 4B= 2Tare hypercyclic, but T, S are clearly not s-hypercyclic. 3. Since each of the operators T, S of the latter example is mixing (see the definition at the beginning of the next section, regarding the sequences of iterates; see also [21, p. 46]), the operator T⊕Sis hypercyclic, but T, S are not s-hypercyclic. 4. De la Rosa and Read [15] were able to construct a Banach space Xand an operator T∈L(X) such that Tis hypercyclic (hence T, T are s-hypercyclic) but Tis not weakly mixing on X, meaning that T⊕Tis not hypercyclic on X2. SIMULTANEOUS UNIVERSALITY 5 While d-hypercyclic operators must be substantially different, s-hypercyclicity allows more similarity. For instance, an operator can never be d-hypercyclic with a scalar multiple of itself (see [11, p. 299]). Nevertheless, s-hypercyclicity is possible in concrete situations. This will be analyzed in Section 4. Sections 5 and 6 are devoted to more specific operators, namely backward shifts and operators on spaces of analytic functions. We close this section by establishing, under appropriate assumptions, the existence of large vector subspaces consisting, except for zero, of s-hypercyclic vectors. Theorem 2.3. (a) Let Xbe a topological vector space and Tj∈L(X) (j= 1,...,p). If T1, . . ., Tpare s-hypercyclic and at least one of them commutes with the others, then s-HC(T1,...,Tp)contains, except for 0, a dense linear subspace of X. (b) Let Xand Ybe two topological vector spaces such that Yis metrizable. Assume that (Tj,n)⊂L(X, Y ) (j= 1,...,p)are hereditarily s-hypercyclic sequences. Then s-HC((T1,n),...,(Tp,n)) contains, except for 0, an infinite dimensional vector subspace of X. (c) Let Xand Ybe two metrizable separable topological vector spaces. Assume that (Tj,n)⊂L(X, Y ) (j= 1,...,p)are hereditarily densely s-hypercyclic sequences. Then s-HC((T1,n),...,(Tp,n)) contains, except for 0, a dense linear subspace of X. Proof. (a) By hypothesis, there is i∈ {1,...,p}such that TiTj=TjTi(j= 1,...,p). Therefore P(Ti)Tj=TjP(Ti) for all jand every polynomial Pwith coefficients in K. Let Pdenote the set of such polynomials. Of course, the operator Tiis hypercyclic. From a result by Wengenroth [29], the operator P(Ti) has dense range as soon as P∈ P \ {0}. Pick any x0∈s-HC(T1,...,Tp). Let us define M:= {P(Ti)x0:P∈ P \ {0}}. Then Mis a linear subspace of X. It is dense because Mcontains the orbit {Tn ix0:n∈N}, that is dense in Xas x0∈HC(Ti). It remains to show that M\ {0} ⊂ s-HC(T1,...,Tp). To this end, fix u∈M\{0}. Then there is P∈ P \{0}such that u=P(Ti)x0. It must be proved that Z⊃∆(Xp), where Z:= {(Tn 1u, . . . , Tn pu) : n∈N}={(P(Ti)Tn 1x0,...,P(Ti)Tn px0) : n∈N}, where the last equality follows from commutativity. We know that ∆(Xp)⊂ {(Tn 1x0,...,Tn px0) : n∈N}. Let A:= {(Tn 1x0,...,Tn px0) : n∈N},ϕ:= P(Ti) and Φ : Xp→Xpbe the mapping defined as Φ(x1,...,xp) := (ϕ(x1),...,ϕ(xp)). Then, as ϕis continuous, we get Z= Φ(A)⊃Φ(A)⊃Φ(∆(Xp)) = {(ϕ(x),...,ϕ(x)) : x∈X}, so Z⊃{(ϕ(x),...,ϕ(x)) : x∈X}. Given y∈Xand a neighborhood Uof (y,y,...,y), there exists a neighborhood Vof ysuch that U⊃Vp. Since ϕ 6 BERNAL AND JUNG has dense range, one can find x∈Xwith ϕ(x)∈V. Then (ϕ(x),...,ϕ(x)) ∈ U. In other words, (y,...,y)∈{(ϕ(x),...,ϕ(x)) : x∈X}, so (y,...,y)∈Z. Consequently, Z⊃∆(Xp), as required. (b)–(c). By mimicking the proofs of Theorems 1–2 of [3] (in which the results are given for a single sequence (Tn)), we can construct recursively a sequence (xN)N∈N⊂Xand a family {(q(N, k))k∈N:N∈N0}of strictly increasing subsequences of Nsatisfying, for all N∈N, the following conditions: xN∈ GN∩s-HC((T1,q(N−1,k),...,(Tp,q(N−1,k))) and Tj,q(l,k)xN→0 as k→ ∞ for all l≥Nand all j∈ {1,...,p}, where G0:= Xand GN:= X\span {x1,...,xN−1} (N∈N) if the assumptions of (b) hold, while {GN}N∈Ndenotes any fixed open basis of Xif the assumptions of (c) hold. Then M:= span {xN:N∈N}is the sought-after vector subspace. The details are left as an exercise.  3. s-Universality criteria A number of workable sufficient conditions will be useful to detect s-universality. Recall that a sequence of continuous mappings Tn:X→Y(n∈N) is called mixing provided that,given nonempty open sets U⊂X,V⊂Y, there is N∈Nsuch that Tn(U)∩V6=∅for all n≥N. The corresponding notion of simultaneous mixing property arises naturally, as well as the one of simultaneous transitivity. Note that Tn(U)∩V6=∅is equivalent to U∩T−1 n(V)6=∅. Definition 3.1. Let p∈Nand X, Y be Hausdorff topological spaces. Assume that, for each j∈ {1,...,p},Tj,n :X→Y(n∈N) is a sequence of continuous mappings. Let also T1,...,Tp:X→Xbe continuous mappings. We say that: (a) The sequences (T1,n),...,(Tp,n) are simultaneously transitive (or s-transitive) provided that, for every pair of nonempty open sets U⊂X,V⊂Y, there is N∈Nsuch that U∩Tp j=1 T−1 j,N (V)6=∅. (b) The sequences (T1,n),...,(Tp,n) are simultaneously mixing (or s-mixing) provided that, for every pair of nonempty open sets U⊂X,V⊂Y, there is N∈Nsuch that U∩Tp j=1 T−1 j,n (V)6=∅for all n≥N. (c) The mappings T1,...,Tpare simultaneously transitive (simultaneously mixing, resp.) whenever the sequences (Tn 1),...,(Tn p) are s-transitive (s-mixing, resp.). Remark 3.2. Corresponding concepts of d-transitivity and d-mixing were introduced in [11], where Tp j=1 T−1 j,n (Vj) (Vjnonempty open subsets of Y,j= 1,...,p) appears instead of Tp j=1 T−1 j,n (V). Also, most criteria given in this section have their counterparts for the related d-properties as provided in [4] and [11]. A thorough study of d-mixing operators is provided in [8]. Note that, contrary to the one-sequence case, the facts U∩Tp j=1 T−1 j,N (V)6=∅ and Tp j=1 Tj,N (U)∩V6=∅are not equivalent. Observe also that Tp j=1 T−1 j,n (V) = SIMULTANEOUS UNIVERSALITY 7 [T1,n,...,Tp,n]−1(Vp). From the definitions, it is easy to check that the sequences (T1,n),...,(Tp,n) are s-mixing if and only if, for every strictly increasing sequence (nk) in N, the sequences (T1,nk),...,(Tp,nk) are s-transitive. The following proposition provides what can be called the Birkhoff s-transitivity theorem. Proposition 3.3. Under the same assumptions and terminology as in Definition 3.1, let us suppose, in addition, that Xis Baire and Yis second-countable. Then we have: (i) The sequences (T1,n),...,(Tp,n)are s-transitive if and only if they are densely s-universal. If this is the case, then the set s-U((T1,n),...,(Tp,n)) is residual in X. (ii) The sequences (T1,n),...,(Tp,n)are s-mixing if and only if, for every strictly increasing sequence (nk)⊂N, the sequences (T1,nk),...,(Tp,nk)are densely s-universal. Proof. Part (ii) is an immediate consequence of (i). Let us prove (i). Fix a countable open basis (Vm) of Y, as well as a point x0∈X. Then x0∈sU((T1,n),...,(Tp,n)) if and only if, given a nonempty open set V⊂Y, there is n∈Nwith [T1,n,...,Tp,n]x0∈Vp, that is, x0∈Sn∈NTp j=1 T−1 j,n (V). Since each V contains some Vmand each Vmis a nonempty subset of Y, the last property is the same as x0∈Tm∈NSn∈NTp j=1 T−1 j,n (Vm), which shows that s-U((T1,n),...,(Tp,n)) = \ m∈N[ n∈N p \ j=1 T−1 j,n (Vm).(1) Since the Tj,n’s are continuous, each set Tp j=1 T−1 j,n (Vm) is open. If (T1,n),...,(Tp,n) are s-transitive then every set Sn∈NTp j=1 T−1 j,n (Vm) (m∈N) is (open and) dense. Hence their (countable) intersection, which equals s-U((T1,n),...,(Tp,n)) by (1), is a dense Gδsubset (so residual) in Xbecause Xis Baire. Conversely, assume that the set of s-universal elements is dense in Xand fix a nonempty open subset Vof Y. Then there is m∈Nwith V⊃Vm. It follows from (1) that Sn∈NTp j=1 T−1 j,n (Vm) is dense in X, so the bigger set Sn∈NTp j=1 T−1 j,n (V) is also dense. But this means that, given a nonempty set U⊂X, there is N∈Nsuch that U∩Tp j=1 T−1 j,N (V)6=∅ or, in other words, the sequences (T1,n),...,(Tp,n) are s-transitive.  In the linear case, we state the following set of sufficient conditions, that are inspired by the results contained in [19, Sect. 1c] and the references cited in it. Theorem 3.4. Let Xand Ybe topological vector spaces such that Xis Baire and Yis metrizable and separable, and let (Tj,n)n∈N(j= 1,...,p)be sequences in L(X, Y ). Assume that there are respective dense subsets X0of Xand Y0of Y satisfying at least one of the following conditions: 8 BERNAL AND JUNG (A) For every pair of vectors x∈X0, y ∈Y0, there exist sequences (nk)⊂N and (xk)⊂Xwith xk→0,Tj,nkx→0and Tj,nkxk→y(j= 1,...,p) as k→ ∞. (B) For every x∈X0, the sequences (Tj,nx)n∈N(j= 1,...,p)converge in Y to a common limit and, for every y∈Y0, there exist sequences (nk)⊂N and (xk)⊂Xwith xk→0and Tj,nkxk→y(j= 1,...,p)as k→ ∞. (C) For every x∈X0, there exists a sequence (nk)⊂Nsuch that the sequences (Tj,nkx)k∈N(j= 1,...,p)converge in Yto a common limit and, for every y∈Y0, there exists a sequence (xn)⊂Xsuch that xn→0and Tj,nxn→y (j= 1,...,p)as n→ ∞. Then (Tj,n)n∈N(j= 1,...,p)are densely s-hypercyclic. Proof. According to Proposition 3.3, we should show that (Tj,n)n∈N(j= 1,...,p) are s-transitive. With this aim, fix a pair of nonempty open sets U⊂X,V⊂Y. We should exhibit an N∈Nsuch that U∩Tp j=1 T−1 j,N (V)6=∅. Assume first that (A) holds. By density, there are x∈X0and y∈Y0such that x∈Uand y∈V. Define A:= U−xand B:= V−y. Then Aand B are open neighborhoods of 0 in Xand Yrespectively. Take a 0-neighborhood C⊂Ysatisfying C+C⊂B. Consider the sequences (nk) and (xk) provided by (A). Then there is k∈Nsuch that xk∈A,Tj,nkx∈Cand Tj,nkxk∈y+C (j= 1,...,p). Let u:= x+xkand N:= nk. We get u∈x+A=Uand Tj,N u=Tj,N x+Tj,N xk∈C+y+C⊂y+B=V(j= 1,...,p), so that u∈U∩Tp j=1 T−1 j,N (V). Suppose now that (B) holds. By density, there is x∈X0such that x∈U. Define A:= U−x, a neighborhood of 0. By hypothesis, there is z∈Ysuch that Tj,n →zas n→ ∞ (j= 1,...,p). Since Y0is dense in Y, there is y∈Y0with y∈z+V. Let B:= V−y+z, a neighborhood of 0 in Y. Take a 0-neighborhood C⊂Ysatisfying C+C⊂B. We have that Tj,n ∈z+C(j= 1,...,p) for n≥n0, say. Consider the sequences (nk) and (xk) provided by (B) for the vector y−z, so that xk→0 and Tj,nkxk→y−z(j= 1,...,p) as k→ ∞. Choose k∈Nso large that nk≥n0,xk∈Aand Tj,nkxk∈y−z+C(j= 1,...,p). Let u:= x+xkand N:= nk. Then u∈x+A=Uand, for every j= 1,...,p, Tj,N u=Tj,N x+Tj,N xk∈z+C+y−z+C=y+C+C⊂y+B=V, so that u∈U∩Tp j=1 T−1 j,N (V), as required. Under assumption (C), the proof is similar and left as an exercise.  Two of the most popular criteria of hypercyclicity are the so-called blow-up/collapse criterion and the hypercyclicity criterion (see [2,20,21]). Now, we can obtain their respective s-versions. Proposition 3.5. [s-Blow-up/Collapse Criterion] Let Xbe a Baire metrizable separable topological vector space, and let (Tj,n)n∈N(j= 1,...,p)be sequences SIMULTANEOUS UNIVERSALITY 9 in L(X). Suppose that, for every nonempty open subsets U, V of Xand every 0-neighborhood W⊂Xthere is N∈Nsuch that W∩ p \ j=1 T−1 j,N (V)6=∅6=U∩ p \ j=1 T−1 j,N (W). Then (Tj,n)n∈N(j= 1, . . . , p)are densely s-hypercyclic. Proof. Fix a pair of nonempty open sets U, V ⊂X. Choose vectors x∈U,y∈V. It suffices to exhibit sequences sequences (nk)⊂Nand (xk)⊂Xwith xk→x and Tj,nkxk→y(j= 1, . . . , p), because this would entail the existence of some k∈Nsuch that xk∈Uand Tj,nkxk∈V(j= 1,...,p), so xk∈U∩Tp j=1 T−1 j,nk(V). In other words, the sequences (Tj,n)n∈N(j= 1,...,p) would be s-transitive, hence densely s-hypercyclic by Proposition 3.3. With this aim, choose a fundamental decreasing sequence (Wk) of 0-neighborhoods. Then (Uk) := (x+Wk) and (Vk) := (y+Wk) are fundamental decreasing sequences of x-neighborhoods and y-neighborhoods, respectively. By hypothesis, for each k∈N, there are nk∈Nand points x′ kand x′′ ksuch that x′ k∈Wk∩ Tp j=1 T−1 j,nk(Vk) and x′′ k∈Uk∩Tp j=1 T−1 j,nk(Wk). Let xk:= x′ k+x′′ k. Then xk→x as k→ ∞ because x′ k∈Wk(so x′ k→0) and x′′ k∈Uk(so x′′ k→x). Finally, Tj,nkxk=Tj,nkx′ k+Tj,nkx′′ k→y+ 0 = y(j= 1,...,p) because Tj,nkx′ k∈Vkand Tj,nkx′′ k∈Wkfor all k∈N. Recall that the convex hull conv(A) of a subset Aof a vector space Xis the least convex subset of Xcontaining A. Definition 3.6. Let Xbe a Baire metrizable separable locally convex space, (nk)⊂Nbe a strictly increasing sequence and Tj∈L(X) (j= 1,...,p). We say that T1,...,Tpsatisfy the s-hypercyclicity criterion with respect to (nk) if there are subsets X0⊂X, W0⊂Xpsuch that X0is dense in Xand W0⊃∆(Xp) as well as mappings Rk:W0→X(k∈N) such that (i) Tnk j→0 pointwise on X0as k→ ∞ (j= 1,...,p), (ii) Rk→0 pointwise on W0as k→ ∞ and (iii) For every w= (w1,...,wp)∈W0and every j∈ {1,...,p}there is yj∈ conv({w1,...,wp}) such that Tnk jRkw→yjas k→ ∞. Theorem 3.7. [s-Hypercyclicity Criterion] Let Xbe a Baire metrizable separable locally convex space and Tj∈L(X) (j= 1,...,p). If T1,...,Tpsatisfy the s-hypercyclicity criterion with respect to some (nk)⊂N, then (Tnk 1),...,(Tnk p)are s-mixing. In particular, T1,...,Tpare densely s-hypercyclic. Proof. Let U, V ⊂Xbe nonempty open sets. Then there are x0∈U∩X0and y0∈V. By local convexity, there is a convex open set e Vwith y0∈e V⊂V. As 16 BERNAL AND JUNG (T2k+1) and ((−T)2k+1) = (−T2k+1) is clearly not possible. In connection with this, it is stated in [7, Remark 24(ii)] and actually proved in [28, Proposition 4.9] that in case of unimodular scalars c1,...,cpevery d-hypercyclic vector x0for T1,...,Tpis also d-hypercyclic for c1T1,...,cpTp. The proof uses crucially the fact that such a vector x0satisfies (x0,...,x0)∈HC(T1⊕ · · · ⊕ Tp). Thus, it cannot be adapted for s-hypercyclicity. Hence, we pose the question: Does the equality s-HC(T1,...,Tp) = s-HC(c1T1,...,cpTp) hold? 3. Concerning again part (b) and regarding its proof, we may obtain a much stronger result in the case K=Cand Xa Banach space. Recall that a nonempty subset E⊂Cis said to be perfect if it is closed and each point of Eis an accumulation point of E. In particular, every perfect set is uncountable. It is well known (see [13, Theorem 8.138(b)]) that there are perfect Dirichlet subsets of T. We have that if E⊂Tis a perfect Dirichlet set and T∈L(X) is mixing, then the uncountable family of rotations {cT :c∈E∪ {1}} is densely uniformly s-hypercyclic, in the sense that there is a dense set of vectors x0∈Xsatisfying the following: for every y∈Xthere is (nk)⊂Nsuch that limk→∞ supc∈E∪{1}k(cT)nkx0−yk= 0. Indeed, we can take a sequence (mk)⊂N such that supc∈E∪{1}|cmk−1|= supc∈E|cmk−1| → 0 as k→ ∞. As Tis mixing, the set HC((Tmk)) is dense. If x0∈HC((Tmk)), then there is a subsequence (nk)⊂(mk) with Tmkx0→y. The conclusion follows from the inequality k(cT)nkx0−yk ≤ kcnk(Tnkx0−y)k+k(cnk−1)yk. 4. Proposition 4.1 furnishes examples of pairs of operators –on spaces of sequences or of holomorphic functions (see sections 5–6)– that are s-hypercyclic but not dhypercyclic: the multiples 2B, −2Bof the backward shift Bon ℓq(1 ≤q < ∞) or c0;D, −Don H(C) (Df := f′); Cϕ,−Cϕon H(G), where Cϕf:= f◦ϕ,G⊂Cis a simply connected domain and ϕis a run-away automorphism of G. 5. Backward shifts and s-hypercyclicity In this section, we consider the sequence spaces c0and ℓq(1 ≤q < ∞) over K=Ror C. If a= (an)n∈Nis a bounded sequence in K\ {0}, then Bawill denote the weighted backward shift Ba: (x0, x1, x2,...)∈X7→ (a1x1, a2x2,...)∈X on X=c0or ℓq. The unweighted backward shift Bis B=Ba, where a= (1,1,1,...). Salas characterized the hypercyclicity of Bain terms of the weight sequence a. B`es and Peris [11, Theorem 4.1] did the same for the d-hypercyclicity of different powers of Ba. This characterization happens to hold also for shypercyclicity. Proposition 5.1. Let X=c0or ℓq(1 ≤q < ∞),p≥2and let r1,...,rp∈N with r1< r2<···< rpbe given. For each l∈ {1,...,p}, let al= (al,n)n∈Nbe a weight sequence. Then the following are equivalent: SIMULTANEOUS UNIVERSALITY 17 (i) Br1 a1,...,Brp apare d-hypercyclic. (ii) Br1 a1,...,Brp apare s-hypercyclic. (iii) For every M > 0and every k∈Nthere is m∈Nsatisfying, for each j∈ {0,1, . . . , k}, that |al,j+1 ···al,j+rlm|> M (1 ≤l≤p)and |al,j+1 ··· al,j+rlm| |as,j+(rl−rs)m+1 ··· as,j+rlm|> M (1 ≤s < l ≤p). (iv) Br1 a1,...,Brp apsatisfy the d-hypercyclicity criterion. (v) Br1 a1,...,Brp apsatisfy the s-hypercyclicity criterion. Proof. The equivalence of (i), (iii) and (iv) is proved in [11, Theorem 4.1]. That (i) implies (ii) is trivial. Moreover, (ii) ⇒(iii) is proved in fact in the proof of “(a) ⇒(b)” of the same reference, since only the simultaneous approximation of one vector (namely e0+···+eq) is used. Finally, we clearly have (iv) ⇒(v) ⇒(ii).  Remarks 5.2. 1. An analogous result about equivalence of dand s-hypercyclicity also works for powers of weighted bilateral shifts (see Theorem 4.7 of [11] and its proof). 2. Corollary 4.4 in [11] also works with just s-universality, as it is a consequence of Theorem 4.1 there. In particular, we have that Ba, B2 a,...,Bp aare s-hypercyclic on Xif and only if Ba⊕B2 a⊕· · ·⊕Bp ais hypercyclic on Xp. B`es, Martin and Peris [7, p. 855] constructed an operator T:= Baon ℓ2such that Tis hypercyclic but T⊕T2is not hypercyclic on ℓ2⊕ℓ2, so that T, T2is not d-hypercyclic on ℓ2. Then we obtain that T, T2are even not s-hypercyclic. According to [21, Theorem 4.8], the mentioned T=Bais not mixing. In [8, Sect. 3], a mixing operator T∈L(ℓ2) for which T, T2are not d-mixing is exhibited. But the existence of a mixing Ton a separable Banach space such that T, T 2are not d-hypercyclic is unknown so far [8, Question 3.7]. A more delicate question arises when r1≤r2≤ · · · ≤ rp. In [11, Corollary 4.2], the following is proved for weighted powers of the unweighted backward shift: if p≥2 and rl∈N,λl∈K(1 ≤l≤p) with r1≤r2≤ · · · ≤ rp, then λ1Br1,...,λpBrpare d-hypercyclic if and only if r1< r2<··· < rpand 1<|λ1|<|λ2|<··· <|λp|. The following result shows that s-hypercyclicity is possible under slightly weaker assumptions. Proposition 5.3. Let p≥2, and let rl∈N,λl∈K(1 ≤l≤p)with r1≤r2≤ · · · ≤ rp. Let Adenote the set A:= {j∈ {1,...,p−1}:rj=rj+1}and consider the conditions (i) 1 <|λj|for all j∈ {1,...,p}, (ii) |λj|<|λj+1|for all j∈ {1,...,p−1} \ A, (iii) |λj|=|λj+1|for all j∈A. Then λ1Br1,...,λpBrpare s-hypercyclic on X=c0or ℓq(1 ≤q < ∞)if and only if (i),(ii) and (iii) hold. 18 BERNAL AND JUNG Proof. First, suppose that conditions (i),(ii) and (iii) hold. We write {1,...,p}\A= {t1,...,td}, with d∈Nand t1<··· < td. As the set {λi/λj:i, j ∈ {1,...,p} with |λi|=|λj|} ⊂ Tis finite, it is a Dirichlet set. Hence there exists a strictly increasing sequence (nk)⊂Nsuch that λi λjnk →1 (k→ ∞) for all i, j ∈ {1,...,p}with |λi|=|λj|.(1) Consider the set X0of finite sequences, that is, X0:= c00 ={x= (xn)∈X: exists n0=n0(x)∈Nsuch that xn= 0 for all n≥n0}. Then X0is dense in X. If we set W0:= ∆(Xp 0)⊂Xp, then W0= ∆(Xp 0)⊃∆(Xp) because X0is dense in X. Now, we set Tj:= λjBrj(j= 1,...,p). Define, for each k∈N, the mapping Rk:W0→Xas follows. If x= (x1, x2,...,xN,0,0,0,...)∈X0and w= (x,x,...,x), then Rkw=(01, u1,02, u2,...,0N, uN,0,0,0,...) if nk≥N (0,0,0,...) if nk< N, where 01:= (0,0, . . . , 0) [rt1nktimes], 0l:= (0,0,...,0) [(rtl−rtl−1)nk−Ntimes] if l≥2 and ul:= 1 λnk tl x1,..., 1 λnk tl xN(l≥1). We have: (a) For each j∈ {1,...,p}and each x= (x1, x2,...,xN,0,0,0,...)∈X0, Tnk jx= 0 as soon as rjnk> N, so Tnk j→0 (k→ ∞) pointwise on X0. (b) For every w= (x,...,x)∈W0as before, the definition of Rktogether with (i) yields Rkw→0 as k→ ∞. (c) Fix w= (x,...,x)∈W0, where x= (x1, x2,...,xN,0,0,0,...). For every j∈ {1,...,p}there is exactly one l∈ {1,...,d}such that |λj|=|λtl|, due to (ii) and (iii). Finally, if nk≥N, we have Tnk jRkw=λj λtlnkx1,λj λtlnkx2,...,λj λtlnkxN,0,0,...,0, λj λtl+1 nkx1,λj λtl+1 nkx2,...,λj λtl+1 nkxN,0,0,...,0,..., λj λtdnkx1,λj λtdnkx2,...,λj λtdnkxN,0,0,0,0,.... It follows from (ii) that ( λj λts)nkxν→0 as k→ ∞ for all s∈ {l+1,...,d}and all ν∈ {1, . . . , N}, while (1) entails that ( λj λtl )nkxν→xνas k→ ∞ for all ν∈ {1,...,N}. Consequently, Tnk jRkw→(x1, x2,...,xN,0,0,0,...) = x. An application of the s-hypercyclicity criterion (see also Remark 3.8.1) concludes the first part of the proof. Now, suppose that λ1Br1,...,λpBrpare s-hypercyclic. Since hypercyclic operators on normed spaces have norm larger than 1, we obtain 1<kλjBrjk=|λj|kBrjk=|λj| SIMULTANEOUS UNIVERSALITY 19 for all j= 1,...,p (cf. the proof of Corollary 4.2 in [11]), i.e. condition (i) holds. For each j∈ {1,...,p−1}\A, we have rj< rj+1. Hence, as λjBrj, λj+1Brj+1 are s-hypercyclic, Proposition 5.1, (ii) ⇒(iii), and the same approach as in the proof of Corollary 4.2 in [11] yield |λj|<|λj+1|, i.e. condition (ii) holds. Finally, for each j∈A, we have rj=rj+1. Hence, the s-hypercyclicity of λjBrj, λj+1Brj+1 =λj+1 λj ·λjBrj implies |λj+1/λj|= 1 (see Proposition 4.1(a)) and thus |λj|=|λj+1|, i.e. condition (iii) holds.  For instance, the operators 2B, 3B2,−3B2, being not d-hypercyclic, are shypercyclic. Further study of d-hypercyclicity of weighted unilateral and bilateral backward shifts can be found in [9]. 6. s-hypercyclicity in spaces of holomorphic functions Let G⊂Cbe a domain, that is, a nonempty connected open subset of C. We endow the space H(G) of all holomorphic (or analytic) functions G→Cwith the topology of uniform convergence on compacta, so that H(G) becomes a separable Fr´echet space. In this section we are concerned with s-hypercyclicity of finite sets of operators on H(G) (or on subspaces of it) for certain domains G. Recall that if Xis a topological vector space and T∈L(X), then Tis said to be supercyclic provided that there exists some x0∈Xwhose projective orbit {λTnx0:n∈N, λ ∈K}is dense in X. If T1,...,Tp∈L(X), they are called dsupercyclic (see [7]) if there is x0∈Xsuch that {λ[Tn 1,...,Tn p]x0:n∈N, λ ∈K} is dense in Xp. Consistently, we say that T1,...,Tpare s-supercyclic whenever {λ[Tn 1,...,Tn p]x0:n∈N, λ ∈K} ⊃ ∆(Xp). Let LFT(D) denote the family of all linear fractional transformations ϕ(z) = az+b cz+dof the complex plane such that ϕ(D)⊂D. The subfamily Aut(D) of automorphisms of Dconsists of all onto members of LFT(D). See e.g. [27, Chapter 1] for terminology related to these families. If ν∈R, then Sνdenotes the weighted Hardy space Sν={f(z) = Pn≥0anzn∈H(D) : kfk:= (Pn≥0|an|2(n+ 1)2ν)1/2<∞}. Each Sνis a Hilbert space, and the choices ν=−1/2,0,1/2 correspond, respectively, to the classical Bergman, Hardy and Dirichlet spaces. Thanks to the results in [7], we obtain without effort the next two assertions. Proposition 6.1. Let ϕ1, . . . , ϕp∈LFT(D)pairwise distinct. Then the following are equivalent: (a) Cϕ1,...,Cϕpare s-supercyclic on H(D). (b) µ1Cϕ1,...,µpCϕpare s-mixing on H(D)for all nonzero scalars µ1,...,µp. (c) Cϕ1,...,Cϕpare d-supercyclic on H(D). (d) µ1Cϕ1,...,µpCϕpare d-mixing on H(D)for all nonzero scalars µ1,...,µp. 20 BERNAL AND JUNG (e) ϕ1...,ϕphave no fixed point in D, and satisfy that if any two ϕl, ϕjhave the same attractive fixed point α, then ϕ′ l(α) = ϕ′ j(α)<1is not possible. Proof. The equivalence of (c), (d) and (e) is proved in [7, Theorem 4]. The implications (d) ⇒(b) ⇒(a) are trivial. Finally, (a) ⇒(e) is proved in fact in the proof of Theorem 4 in [7]. Indeed, it is used there a result (Lemma 14 in [7]) asserting that if ϕ1, ϕ2∈LFT(D) are hyperbolic and share an attractive fixed point αwith ϕ′ 1(α) = ϕ′ 2(α), then Cϕ1, Cϕ2are not d-supercyclic on H(D). But a closer look at its proof shows that Cϕ1, Cϕ2are in fact even not s-supercyclic; indeed, via contradiction, only one function gis assumed to be simultaneously approximated by projective orbits.  Proposition 6.2. Let ϕ1,...,ϕp∈LFT (D)pairwise distinct and let ν < 1/2. Then the following are equivalent: (a) Cϕ1,...,Cϕpare s-supercyclic on Sν. (b) Cϕ1,...,Cϕpare s-mixing on Sν. (c) Cϕ1,...,Cϕpare d-supercyclic on Sν. (d) Cϕ1,...,Cϕpare d-mixing on Sν. (e) Each ϕlis a parabolic automorphism or a hyperbolic map without fixed points in D, and there are no two ϕl, ϕjhaving a common fixed point α such that ϕ′ l(α) = ϕ′ j(α)<1. Proof. The equivalence of (c), (d) and (e) is proved in [7, Theorem 3]. The implications (d) ⇒(b) ⇒(a) are trivial. As for (a) ⇒(e), observe that in the proof of Theorem 3 in [7], only the supercyclicity of each Cϕlis necessary for the first assertion in (e) and that the Comparison Principle [7, Proposition 8] –that also works for s-supercyclicity– implies that Cϕ1,...,Cϕpare s-supercyclic on H(D). Now, the second assertion of (e) follows from Proposition 6.1.  Remarks 6.3. 1. Recall that if Xis an F-space and T∈L(X) is invertible and hypercyclic, then T−1is also hypercyclic. Analogously as in Example 22 in [7], by combining the preceding two propositions, we obtain that there are hyperbolic ϕ1, ϕ2∈Aut(D) such that Cϕ1, Cϕ2are d-hypercyclic (so s-hypercyclic) on H2(D) (the Hardy space) and on H(D), and Cϕ−1 1= (Cϕ1)−1, Cϕ−1 2= (Cϕ2)−1 are even not s-supercyclic on H2(D) or H(D) (note that ϕ−1 1and ϕ−1 2are also hyperbolic). Hence, in general, the d-hypercyclicity of T1,...,Tpdoes not imply the s-hypercyclicity of T−1 1,...,T−1 pif T1,...,Tpare invertible. Moreover, finitely many composition operators generated by non-elliptic automorphisms of Dmay be not s-hypercyclic on H(D) or on H2(D). 2. Further study of d-hypercyclicity of composition operators, this time on weighted Bergman spaces on D, is performed in [30]. In 1929 Birkhoff [12] proved that the translation operator τa(a∈C\ {0}) given by (τaf)(z) = f(z+a) is hypercyclic on the space H(C) of entire functions. It is SIMULTANEOUS UNIVERSALITY 21 proved in [4, Prop. 5.5] and [11, Theorem 3.1] that if a1,...,apare pairwise distinct nonzero complex numbers, then τa1,...,τapare d-hypercyclic. Trivially, we obtain: if a1,...,ap∈C\ {0}, then τa1,...,τapare s-hypercyclic. As the next proposition shows, we may obtain a slight extension to weighted translation operators. Proposition 6.4. Let p≥2, and let a1,...,ap, λ1,...,λp∈C\ {0}such that |λj|=|λl|for all j, l ∈ {1,...,p}with aj=al. Then there is a sequence (nk)⊂N such that the sequences (λ1τa1)nk,...,(λpτap)nkare s-mixing. In particular, the operators λ1τa1,...,λpτapare densely s-hypercyclic on H(C). Proof. Select a finite sequence {j(1) < j(2) ··· < j(q)} ⊂ {1,...,p}satisfying that, if bl:= aj(l)(l= 1,...,q), then the bl’s are pairwise distinct and {a1,...,ap}={b1,...,bq}. Let µl:= λj(l). Consider the operators Tj:= λjτaj (j= 1,...,p) and Sl:= Tj(l)=µlτbl(l= 1,...,q). Let us prove that S1, . . . , Sqare s-mixing. In fact, by following the approach of the proof of [11, Theorem 3.1], we can prove that they are even d-mixing. To this end, and taking into account that the sets V(h, r, ε) := {f∈H(C) : |f(z)−h(z)|< ε for all z∈B(0, r)}(h∈H(C), ε > 0, r > 0), form a basis for the topology of H(C), it is enough to prove that, for given h, g1,...,gq∈H(C) and ε, r > 0, there is n0∈Nsuch that, for every n≥n0, there exists an entire function fwith |f(z)−h(z)|< ε and |(Sn lf)(z)−gl(z)|< ε (z∈B(0, r), l = 1,...,q).(1) Select n0∈Nwith n0>maxi6=l2r |bi−bl|+ max1≤l≤q2r |bl|. Then, for each n≥n0, the disks B(0, r), B(nb1, r), . . . , B(nbq, r) are pairwise disjoint. Pick s > r such that the disks B(0, s), B(nb1, s), . . . , B(nbq, s) are still pairwise disjoint. Let K:= B(0, r)∪B(nb1, r)∪ · · · ∪ B(nbq, r) and Ω := B(0, s)∪B(nb1, s)∪ · · · ∪ B(nbq, s). Note that Ω is an open set, Ω ⊃Kand Kis a compact subset having connected complement. Consider the function F: Ω →Cdefined by F(z) = h(z) if z∈B(0, s) and F(z) := µ−n lgl(z−nbl) if z∈B(nbl, s) (1 ≤l≤q). Then F∈H(Ω). From Runge’s approximation theorem (see e.g. [16]), it follows that there exists a polynomial f(so f∈H(C)) such that |f(z)−F(z)|< ε/(1 + |µn l|) for all z∈K. But this implies that |f(z)−h(z)|< ε on B(0, r) and |µn lf(z)−gl(z−nbl)|< ε on B(nbl, r). Since the last inequality is equivalent to |µn lf(z+nbl)−gl(z)|< ε on B(0, r), (1) is obtained. As the set D:= {λj/λl:j, l ∈ {1,...,p}with aj=al} ⊂ Tis finite, it is a Dirichlet set. Then there is a strictly increasing sequence (nk)⊂Nsuch that ξnk→1 as k→ ∞, for all ξ∈D. Fix a subsequence (mk) of (nk). Since S1,...,Sqare s-mixing, the set s-HC((Smk 1),...,(Smk q)) is dense (see Proposition 3.3). Fix fin s-HC((Smk 1),...,(Smk q)). For each ν∈ {1,...,p}there is a unique l=l(ν)∈ 22 BERNAL AND JUNG {1,...,q}such that aν=bl, so that |λν|=|µl|. Observe that ξν:= λν/µl∈D. Then ξnk ν→1, hence ξmk ν→1 (k→ ∞) for all ν∈ {1,...,p}. Given g∈H(C), we can find a subsequence (pk) of (mk) with Spk l(ν)f→g(k→ ∞) uniformly on compacta for every ν∈ {1,...,p}. Since ξpk ν→1 for all ν, we obtain that Tpk νf=ξpk νSpk l(ν)f−→ 1·g=g(k→ ∞) uniformly on compacta for every ν= 1,...,p. Therefore f∈s-HC((Tmk 1),...,(Tmk p)), which shows that this set is dense. By Proposition 3.3, the sequences (Tnk 1),...,(Tnk p) are s-mixing, as required.  Another important collection of operators on H(C) is that of differentiation operators. Consider the derivative operator D:f∈H(C)7→ f′∈H(C). Its hypercyclicity on H(C) was proved by MacLane in 1952 [25]. It is shown in [11, Prop. 3.3] that if p≥2, r1,...,rp∈Nwith r1<··· < rpand λ1,...,λp∈ C\{0}, then λ1Dr1,...,λpDrpare d-mixing, so densely d-hypercyclic. Concerning s-hypercyclicity, the following proposition shows that somewhat softer assumptions are allowed, although, similarly to the last proposition, we have not been able to obtain the s-mixing property for the whole sequences. Proposition 6.5. Let r1≤ · · · ≤ rpbe positive integers and λ1,...,λp∈C\{0}, where p≥2. Suppose that |λj|=|λl|for all j, l ∈ {1,...,p}with rj=rl. Then there is a sequence (nk)⊂Nsuch that the sequences (λ1Dr1)nk,...,(λpDrp)nk are s-mixing. In particular, the operators λ1Dr1,...,λpDrpare densely s-hypercyclic on H(C). Proof. As the set {λj/λl:j, l ∈ {1,...,p}with rj=rl} ⊂ Tis finite, it is a Dirichlet set. Then there is a strictly increasing sequence (nk)⊂Nsuch that (λj/λl)nk→1 as k→ ∞, for all j, l ∈ {1,...,p}with rj=rl. Put X0:= {polynomials}= span{zm:m∈N0}and W0:= ∆(Xp 0). Then X0is dense in X:= H(C) and W0= ∆(Xp 0)⊃∆(Xp). Let Tj:= λjDrj(1 ≤j≤p). For each k∈N, define the map Rk:W0→Xvia Rk(zm, . . . , zm) := p X l=1 1 τ(l)·1 λnk l ·zm+rlnk (m+ 1)(m+ 2) ···(m+rlnk), where τ(l) := card {i∈ {1,...,p}:ri=rl}(1 ≤l≤p). Then Rkis extended to the whole W0by linearity. We have: (i) Tnk jzm= 0 as soon as nkrj> m, so Tnk jzm→0 as k→ ∞ for all j∈ {1,...,p}and all m≥0. Therefore, by linearity, Tnk j→0 (k→ ∞) on X0 for all j∈ {1,...,p}. SIMULTANEOUS UNIVERSALITY 23 (ii) Fix m∈N0and a compact set K⊂C. There is M∈(0,+∞) with K⊂B(0, M). Given k∈N, we obtain sup z∈K |Rk(zm,...,zm)| ≤ p X l=1 1 τ(l)·1 λnk l ·Mm+rlnk (m+ 1)(m+ 2) ···(m+rlnk) ≤ p X l=1 1 τ(l) Mm+rlnk/λnk l (m+ 1)(m+ 2) ···(m+nk) ≤ p X l=1 Mm τ(l) (Mrl/λl)nk nk!→0 (k→ ∞) Hence, by linearity, Rk→0 (k→ ∞) pointwise on W0. (iii) Fix m∈N0,j∈ {1,...,p}and k∈Nwith nk> m. Let us compute the action of Tnk jRkon each (zm,...,zm). This yields three sums, the first of them corresponding to those l∈ {1,...,p}with rl< rj, that equals 0. Therefore Tnk jRk(zm,...,zm) = 0 + p X l=1 rl=rj 1 τ(l)·λj λlnk·zm + p X l=1 rl>rj 1 τ(l)·λj λlnk·zm+(rl−rj)nk (m+ 1)(m+ 2) ···(m+ (rl−rj)nk) −→ 1 τ(j)·zm· p X l=1 rl=rj 1 + 0 = zm(k→ ∞) uniformly on compacta in C, because τ(j) = τ(l) and (λj λl)nk→1 for all (j, l) with rj=rl. By linearity again, we get Tnk jRk(w,...,w)→wfor all j= 1,...,p and all (w,...,w)∈W0. The conclusion now follows from Theorem 3.7 (or from Remark 3.8.1).  For instance, the operators 5D, D2,−D2, eiD2,1 10D3,−3D4are s-hypercyclic, but clearly not d-hypercyclic. An extension unifying both Birkhoff’s and MacLane’s theorems takes place by considering convolution operators on H(C), that is, operators commuting with all translations τa. Let Φ(z) = P∞ n=0 anzn∈H(C). Then Φ is said to be of exponential type provided that there are positive constants A, B such that |Φ(z)| ≤ Aexp(B|z|) for all z∈C. Then its associated differential operator Φ(D) = P∞ n=0 anDngiven by Φ(D)f=P∞ n=0 anf(n)(f∈H(C)) defines an operator on H(C). Moreover, an operator T∈L(H(C)) is of convolution if and only if T= Φ(D) for some entire function Φ of exponential type. Note that Dand τa 24 BERNAL AND JUNG are special cases (take Φ(z)≡zand Φ(z)≡eaz, resp.). Godefroy and Shapiro [17] proved in 1991 that any nonscalar convolution operator is hypercyclic. If G is any domain in C, then Φ(D) is also an operator on H(G) whenever Φ is of subexponential type, that is, for given ε > 0 there is a constant A > 0 such that |Φ(z)| ≤ Aexp(ε|z|) for all z∈C. We have that also Φ(D) is hypercyclic on H(G) provided that Gis simply connected (i.e. its complement with respect to the one-point compactification C∞of Cis connected) and Φ is not constant. For s-hypercyclicity, we present the following assertion, with which we put an end to this introductory paper on s-universality. Proposition 6.6. Assume that G⊂Cis a simply connected domain and that Φ1,...,Φpare entire functions of subexponential type (or just of exponential type if G=C). Assume also that the set U0:= λ∈C: max 1≤j≤p|Φj(λ)|<1 is nonempty and that each set Ui:= λ∈C:|Φi(λ)|>1 and max 1≤j≤p|Φj(λ)| ≤ |Φi(λ)|(1 ≤i≤p) has nonempty interior U0 i. Suppose, in addition, that whenever i, j ∈ {1,...,p} satisfy |Φi(λ)|=|Φj(λ)|for some λ∈U0 i, there exists ζ∈Twith Φj=ζ·Φi. Then there is a sequence (nk)⊂Nsuch that the sequences (Φ1(D))nk,... . . . , (Φp(D))nkare s-mixing. In particular, the operators Φ1(D),...,Φp(D)are densely s-hypercyclic on H(C). Proof. We write eλ:= exp(·λ)|Gfor λ∈C. It is easy to see that the functions eλ are linearly independent. Denote Vi:= U0 i(1 ≤i≤p). As U0, V1,...,Vpare open and nonempty, we obtain that X0:= span{eλ:λ∈U0}is dense in X:= H(G) (because Gis simply connected: use Runge’s approximation theorem together with the fact that span{exp(·λ) : λ∈U0}is dense in H(C); see e.g. [17, Sect. 5]). Hence W0:= Qp i=1 span{eλ:λ∈Vi}is dense in Xp. As A:= {ζ∈T: exist l, j ∈ {1,...,p}with Φj=ζΦl} ⊂ Tis finite, it is a Dirichlet set; hence there is a strictly increasing sequence (nk)⊂Nsuch that ζnk→1 for all ζ∈A. For each i∈ {1, . . . , p}, we put Ti:= Φi(D)|H(G),Ei:= {j∈ {1,...,p}: exists ζ∈Twith Φj=ζΦi}and τ(i) := card(Ei). Notice that if i∈Ej, then Ei=Ej (just use that Tis a multiplicative group), hence τ(i) = τ(j). Given i∈ {1,...,p} and vi∈span{eλ:λ∈Vi}, there are uniquely determined scalars ci,1,...,ci,J(i)∈ Cand pairwise distinct λi,1,...,λi,J(i)∈Visuch that vi=PJ(i) l=1 ci,leλi,l . For k∈N we define Rk:W0→Xas Rkw:= p X i=1 1 τ(i)· J(i) X l=1 ci,l Φi(λi,l)nk·eλi,l ,(1) SIMULTANEOUS UNIVERSALITY 25 where w= (v1,...,vp)∈W0and the vi’s are as above. We have: (i) If λ∈U0and j∈ {1,...,p}, then Tnk jeλ= Φj(λ)nkeλ→0 as k→ ∞, because |Φj(λ)|<1. By linearity, we get Tnk j→0 on X0. (ii) Let w= (v1,...,vp)∈W0, so that vi=PJ(i) l=1 ci,leλi,l , as above. Since |Φi(λi,l)|>1, we get |Φi(λi,l)nk| → +∞as k→ ∞, for each i∈ {1,...,p} and each l= 1, . . . , J(i). From (1) one derives that Rkw→0. (iii) Again, let w= (v1,...,vp)∈W0, with vi=PJ(i) l=1 ci,leλi,l . Fix j∈ {1,...,p}and k∈N. We compute Tnk jRkw= p X i=1 1 τ(i)· J(i) X l=1 ci,l Φi(λi,l)nk·Tnk jeλi,l = p X i=1 1 τ(i)· J(i) X l=1 ci,l ·Φj(λi,l) Φi(λi,l)nk eλi,l =Ak+Bk, where Ak(Bk, resp.) denotes the part of the preceding sum corresponding to those i∈Ej(i6∈ Ej, resp.). If i∈Ej, there is ζ=ζi,j ∈Asuch that Φj=ζ·Φi, so that Φj(λi,l) Φi(λi,l)nk=ζnk→1 as k→ ∞. Note that τ(i) = τ(j) if i∈Ej. Therefore, on the one hand, Ak→ p X i=1 i∈Ej 1 τ(i)· J(i) X l=1 ci,l ·eλi,l =1 τ(j)· p X i=1 i∈Ej J(i) X l=1 ci,l ·eλi,l =1 τ(j)· p X i=1 i∈Ej vi. On the other hand, if i6∈ Ej, we have that |Φj(λi,l)/Φi(λi,l)|<1 for all l= 1,...,J(i) (indeed, as λi,l ∈Vi, we have |Φj(λi,l)| ≤ |Φi(λi,l)|; if we assume |Φj(λi,l)|=|Φi(λi,l)|, then there would exist ζ∈Twith Φj=ζ·Φi, which would yield i∈Ej, a contradiction). Hence Φj(λi,l) Φi(λi,l)nk→0, so Bk→0. This entails Tnk jRkw=Ak+Bk→1 τ(j)· p X i=1 i∈Ej vi(k→ ∞), and the last vector belongs to conv({v1,...,vp}) since in the last sum there are exactly τ(j) summands. The conclusion follows, once again, from the s-hypercyclicity criterion (Theorem 3.7).  Remark 6.7. Proposition 3.4 in [11] (see also [4, Theorem 5.3]) asserts that if U0 and Wi:= {λ∈C:|Φi(λ)|>1 and maxj6=i|Φj(λ)|<|Φi(λ)|} (1 ≤i≤p) are nonempty, then Φ1(D),...,Φp(D) are d-mixing. If these assumptions are satisfied, then the assumptions of Proposition 6.6 are also satisfied. Note that Proposition 6.6 includes the case Φ1= Φ, Φj=cjΦ with |cj|= 1 (j= 2,...,p).