# On the RecurrentC0-Semigroups, Their Existence, and Some Criteria

Authors: Mansooreh Moosapoor
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, recurrentC0-semigroups are introduced and investigated. It is proved that, despite hypercyclic C0-semigroups, recurrent C0-semigroups can be found on finite-dimensional Banach spaces. Some criteria are stated for recurrence, which is based on open sets, neighborhoods of zero, and special eigenvectors. It is established that having a dense set of recurrent vectors is a sufficient and necessary condition for a C0-semigroup to be recurrent. Moreover, the direct sum of recurrent C0-semigroups is investigated.


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## Body

## 1. Introduction

The study of the dynamical system is a notable branch in mathematics. Hypercyclicity, recurrency, chaoticity, and mixing are investigated in this branch. For a Banach spaceX, an operator is named hypercyclic if orbT,x¯=X for some x∈X or equivalently T−nU∩V≠ϕ for some integer n≥1, where U and V are arbitrary open sets in X. If for any open set U in X, a positive integer n can be chosen such that T−nU∩U≠ϕ; then, T is called a recurrent operator. Recurrence is a remarkable case of hypercyclicity. In fact, in this case, the inverse image of any open set under the operator intersects with itself. More information are accessible in [1, 2].A hypercyclic operatorT on X, with a dense set of periodic points is named a chaotic operator. Also, T is called mixing if for any open set U and V of X, TnU∩V≠ϕ for any n greater or equal a natural number N. References [3, 4] contain valuable information about the above notions and results.One of the significant structures that are considered by mathematicians isC0-semigroups. Suppose that Ttt≥0 be a family of operators. Ttt≥0 is called a C0-semigroup if for any sandt≥0, T0=I, Ts+t=TsTt, and lims⟶tTsx=Ttx, for any x∈X.Similar to operators, hypercyclicC0-semigroups, chaotic C0-semigroups, and mixing C0-semigroups are defined.AC0-semigroup Ttt≥0 on X is named hypercyclic if it has a dense orbit or equivalently for any open sets U and V of X, there is t>0 such that Tt−1U∩V≠ϕ. One can see [5] either. Moreover, there are remarkable criteria in this matter that can be found in [6]. Also, new points about constructive approximation of semigroups can be observed in [7]. A hypercyclic C0-semigroup Ttt≥0 is called chaotic if it has a dense set of periodic points. It means there are points like x∈X with this property that Ttx=x for some t>0. Also, a C0-semigroup Ttt≥0 on X is called mixing if for any open sets U and V of X, s>0 can be chosen such that, for any t≥s, TtU∩V≠ϕ.In the following, we mention hypercyclicity criterion and a recurrent hypercyclicity criterion forC0-semigroups.Definition 1.
(see [8])(HCC). A semigroup Ttt≥0 on X fulfills the hypercyclicity criterion if and only if t>0 can be found such that(1)TtU∩W≠ϕ,TtW∩V≠ϕ,where U and V are arbitrary nonempty and open sets and W is an arbitrary neighborhood of zero in X.Definition 2.
(see [8])(RHCC). A semigroup Ttt≥0 on X fulfills the recurrent hypercyclicity criterion if and only if for any nonempty and open sets U and V and any neighborhood W1 of zero in X, L1>0, and L2>0 can be found such that, for any t≥0, s1∈J∩t,t+L1 and s2∈J∩t,t+L2 can be found such that(2)Ts1U∩W≠ϕ,Ts2W∩V≠ϕ.
Desch and Schappacher [8] proved that if a semigroup is RHCC, then it is HCC. In [9, 10], more matters can be found about properties of C0-semigroups.
By various types ofC0-semigroups and their notable properties, it sounds interesting to investigate C0-semigroups with this property that returns back an open set to itself. If we name this new concept as recurrent C0-semigroup, the question arises as to whether we can find interesting properties for them or whether we can get some relations between them and concepts such as hypercyclicity, chaoticity, and mixing for C0-semigroups?
In this paper, we introduce the concept of recurrentC0-semigroups and look into their properties. In Section 2, we state some preliminaries about this concept. We show that the recurrence of any operator in a C0-semigroup implies the C0-semigroup recurrence. Also, if Ttt≥0 satisfies HCC, then the recurrence of Tt, can be concluded for any t>0.
In Section3, recurrent vectors for C0-semigroups are defined. It is proved that a C0-semigroup Ttt≥0 on X is recurrent if and only if RecTtt≥0¯=X. It is proved that recurrence preserves under conjugacy. We establish that, despite hypercyclic C0-semigroups, recurrent C0-semigroups can be constructed on finite-dimensional spaces. Also, we make various examples of recurrent C0-semigroups.
In Section4, various criteria for recurrence of C0-semigroups are presented. The conditions of these criteria are based on open sets, neighborhoods of zero, dense sets, and special eigenvectors.
In Section5, we investigate the recurrence of the direct sum of two C0-semigroups. We establish that recurrence of the direct sum of two semigroups implies recurrence of each of them. Moreover, if at least one of them be mixing, the converse is also true. Additionally, it is proved that if one of the C0-semigroups is RHCC and the other C0-semigroup is HCC, then their direct sum is recurrent.


## 2. Preliminaries

We begin this section by defining the concept of recurrentC0-semigroup.Definition 3.
We say aC0-semigroup Ttt≥0 is recurrent if for any open and nonempty set U, some t>0 can be found such that Tt−1U∩U≠ϕ.
By Definition3, it is not complicated to see that if Ttt≥0 is a hypercyclic C0-semigroup or a chaotic C0-semigroup, then Ttt≥0 is recurrent. Moreover, we establish that hypercyclicity (chaoticity) of an operator in a C0-semigroup implies its hypercyclicity (chaoticity) as follows.Corollary 1.
Suppose thatTtt≥0 be a C0-semigroup on X. If Ts be hypercyclic or chaotic for some s>0, then Ttt≥0 is recurrent.Proof.
Hypercyclicity (chaoticity) ofTs for some s>0 indicates that Ttt≥0 is hypercyclic (chaotic). Accordingly, Ttt≥0 is recurrent.
The next theorem shows that the recurrence of any operator in aC0-semigroup implies its recurrence.Theorem 1.
ConsumeTtt≥0 is a C0-semigroup on X. Then, recurrence of Ttt≥0 is derived from recurrence of any of Tt’s.Proof.
LetTt0 be recurrent for some t0>0. Consider U be an open and nonempty set. Hence, there is n∈ℕ such that Tt0−nU∩U≠ϕ. Suppose x∈Tt0−nU∩U. Then, Tt0nx∈U and x∈U. But Ttt≥0 is a C0-semigroup. Hence, Tt0n=Tnt0. So, Tnt0x∈U. Therefore, x∈Tnt0−1U∩U. Thus, Ttt≥0 is recurrent.
Now, this question arises that if the converse of Theorem1 is true?
The next theorem manifests that the answer is affirmative ifTtt≥0 fulfills the hypercyclicity criterion.Theorem 2.
IfTtt≥0 fulfills the hypercyclicity criterion, then Tt is a recurrent operator for any t≥0.Proof.
LetTtt≥0 fulfills the hypercyclicity criterion. Thus, Ttt≥0 is hypercyclic (Theorem 7.27 in [4]). Hence, Tt is hypercyclic, and so, it is recurrent for any t≥0 (Theorem 2.3 in [11]).
The following lemma states that, for aC0-semigroup Ttt≥0 and any open and nonempty set U, Tt−1U∩U≠ϕ for infinitely many t>0.Lemma 1.
LetTtt≥0 be a recurrent C0-semigroup. Then,(3)t>0:Tt−1U∩U≠ϕ,which is an infinite subset of ℝ+.Proof.
Assume, on the contrary, there is an open and nonempty setU such that(4)t>0:Tt−1U∩U≠ϕ=t1,t2,…,tn.
Suppose thattn be the greatest number in the right set of (4). Now, Ttn−1U∩U is nonempty and open. So, there is ts>0 such that(5)Tts−1Ttn−1U∩U∩Ttn−1U∩U≠ϕ.
Hence,Ttn+ts−1U∩U≠ϕ. That means tn+ts belongs to the left set of (4). But this is a contradiction since tn+ts>tn.


## 3. Recurrent Vectors, Frequently Recurrent Vectors, and Finite Dimensions

A vectorx∈X is named a recurrent vector for an operator T if Tnkx⟶x, where nk is an increasing sequence. By this notion, we can state the next lemma.Lemma 2.
Suppose thatTtt≥0 is a C0-semigroup on X. Then, Ttt≥0 is recurrent if for some t0>0, RecTt0 be dense in X.Proof.
Suppose thatRecTt0¯=X for some t0>0. Then, Tt0 is recurrent (Proposition 2.1 in [1]). Now, by Theorem 1, Ttt≥0 is recurrent.
The concept of the recurrent vector can be defined forC0-semigroups as follows.Definition 4.
A vectorx is named a recurrent vector for Ttt≥0 if Ttnx⟶x for some increasing sequence tn. We denote the set of recurrent vectors of Ttt≥0 by RecTtt≥0.
It is affirmed in the next theorem that having a dense set of recurrent vectors is an equivalent condition for aC0-semigroup to be recurrent.Theorem 3.
LetTtt≥0 be a C0-semigroup on X. Then, Ttt≥0 is recurrent if and only if RecTtt≥0¯=X.Proof.
LetRecTtt≥0¯=X. Suppose that U be an open and nonempty subset of X. So, x∈RecTtt≥0∩U can be found. Hence, there is an increasing sequence tn such that Ttnx⟶x. Therefore, n0∈ℕ can be chosen such that Ttn0x∈U. Hence, x∈Ttn0−1U∩U.
Now, suppose thatTtt≥0x∈X. Consider U=Bx,ϵ, where ϵ<1. By recurrence of Ttt≥0, t0>0 can be found such that Tt0−1U∩U≠ϕ. So, there is x1∈Tt0−1U∩U.
Letϵ1<1/2 be such that(6)U1=Bx1,ϵ1⊆Tt0−1U∩U.
Another by recurrence,t1>t0 can be found such that Tt1−1U1∩U1≠ϕ. Therefore, x2∈X and ϵ2<1/22 can be chosen such that x2∈Tt1−1U1∩U1 and(7)U2=Bx2,ϵ2⊆Tt1−1U1∩U1.
Inductively, an increasing sequencetn can be found such that, for any n≥1,(8)Un=Bxn,ϵn⊆Ttn−1−1Un−1∩Un−1.
So, for anyn≥1,(9)Ttn−1Bxn,ϵn⊆Un−1.
Now, the Cantor theorem implies that∩n=1∞Un=z for some z∈X. Also, Ttnz⟶z by (9). Hence, z is a recurrent vector. Moreover, x−z<ϵ.
Assume thatTtt≥0 and Stt≥0 be two C0-semigroups on spaces X and Y, respectively. If a continuous map φ:Y⟶X can be found with this property that Tt∘φ=φ∘St for any t≥0, then Ttt≥0 and Stt≥0 are named quasi-conjugate. We state in the following theorem that quasi-conjugacy preserves recurrence.Theorem 4.
Recurrence ofC0-semigroups preserves under quasi-conjugacy.Proof.
LetStt≥0 be a recurrent C0-semigroup, and let Ttt≥0 and Stt≥0 be quasi-conjugate. Let U be an open and nonempty set. Now, φ−1U is open by continuity of φ. Since Stt≥0 has a dense set of recurrent vectors, there exists x∈RecStt≥0 such that x∈φ−1U. Hence, n0∈ℕ can be found such that Stn0x∈φ−1U. So, there is n0>0 such that φ∘Stn0x∈U. Now, by conjugacy, φ∘Stn0=Ttn0∘φ, and thus, we can conclude that Ttn0∘φx∈U. Hence, φx∈Ttn0−1U∩U. Therefore, Ttt≥0 is recurrent.
Pay attention to the fact that periodic points of aC0-semigroup are recurrent. In fact, if x be a periodic point for C0-semigroup Ttt≥0, then Tt0x=x for some t0>0. Hence, for any n∈ℕ, we have Tnt0x=x. Therefore, Tnt0x⟶x, and hence, x is a recurrent vector for Ttt≥0.
By this fact, interesting examples can be constructed as follows.Example 1.
Suppose thatTtt≥0 be a C0-semigroup on T so that, for any t≥0, Tt:T⟶T is defined by Ttx=eitx. Let t0=2/Nπ. So, Tt0N=I, and hence, every point of T is a periodic point for Tt0. Hence, the set of periodic points of Tt0 is dense, and hence, Tt0 is recurrent. So, by Theorem 3, Ttt≥0 is recurrent.
Bermúdez et al. in [12] proved that there are hypercyclic C0-semigroups that are not chaotic. Hence, there exist recurrent C0-semigroups that do not have a dense set of periodic points.
By Theorem 2.4 in [12], hypercyclic C0-semigroups can be found on any infinite-dimensional and separable Banach spaces. So, we can deduce that recurrent C0-semigroups exist on these spaces since hypercyclic C0-semigroups are recurrent. By Example 1, we can deduce the following theorem.Theorem 5.
There are finite-dimensional Banach spaces that support recurrentC0-semigroups.Proof.
In Example1, we make a recurrent C0-semigroup on a finite-dimensional Banach space. So, recurrent C0-semigroups can be found on finite-dimensional spaces.
As it is established in Theorem 7.15 in [4], hypercyclic C0-semigroups cannot be built on finite-dimensional spaces. So, we can state the following corollary.Corollary 2.
The set of hypercyclicC0-semigroups is a proper subset of the set of recurrent C0-semigroups.Proof.
The proof is evident by Theorem5 and this fact that hypercyclic C0-semigroups do not exist on finite-dimensional spaces.
Frequently recurrent vectors for operators are defined in [13]. Similarly, frequently recurrent vectors for C0-semigroups can be defined as follows.Definition 5.
A vectorx∈X is named a frequently recurrent vector for a C0-semigroup Ttt≥0 on X if(10)∪t≥0n≥0:Tntx∈U,has positive lower density for any open set U that contains x. We denote the set of frequently recurrent vectors of Ttt≥0 by FRecTtt≥0.
By Definition5, it is not complicated to see that(11)PerTtt≥0⊆FRecTtt≥0⊆RecTtt≥0.
By Theorem3 and (11), we can conclude the following corollary.Corollary 3.
If aC0-semigroup Ttt≥0 has a dense set of frequently recurrent vectors, then it is recurrent.In the next example, aC0-semigroup is constructed with a dense set of frequently recurrent vectors.Example 2.
LetX=C0ℝ+, where(12)C0ℝ+=f:f:ℝ+⟶ℂ,limx⟶∞fx=0.
Considerf=supx∈ℝ+fx. Let β>0. If for any f∈C0ℝ+ and x∈ℝ+ we define(13)Ttfx=eβtfx+t,then Ttt≥0 is a chaotic C0-semigroup (Example 7.10 in [4]). Hence, it has a dense set of periodic points in X. Therefore, Ttt≥0 has a dense set of frequently recurrent vectors.


## 4. Some Criteria for Recurrence ofC0-Semigroups

Some criteria for recurrency ofC0-semigroups are presented in this section. The conditions of the first theorem are based on open sets and neighborhoods of zero. Moreover, it has weaker conditions than conditions in HCC.Theorem 6.
AssumeTtt≥0 be a C0-semigroup on X. If for any open and nonempty set U and any neighborhood W of zero, there is some t>0 such that(14)TtU∩W≠ϕ,TtW∩U≠ϕ,and then, Ttt≥0 is recurrent.Proof.
Suppose thatU be an open and nonempty set. So, there is an open set U1 and a neighborhood W1 of zero so that U1+W1⊆U (Lemma 2.36 in [4]). By hypothesis, there is t>0 such that(15)TtU1∩W1≠ϕ,TtW1∩U1≠ϕ.
Hence, there areu∈U1 and w∈W1 such that(16)Ttu∈W1,Ttw∈U1.
Now, by (16),(17)Ttu+w=Ttu+Ttw∈U,u+w∈U.
Therefore,u+w∈U and Ttu+w∈U. Hence, t>0 can be selected such that Tt−1U∩U≠ϕ. So, Ttt≥0 is recurrent.
We can rewrite Theorem6 as follows.Theorem 7.
ConsiderTtt≥0 be a C0-semigroup on X. If there is α>0 such that, for any open and nonempty set U and any neighborhood W of zero, t>0 can be found so that(18)TtU∩W≠ϕ,Tt+αW∩U≠ϕ,and then, Ttt≥0 is recurrent.Proof.
Assume thatW is a neighborhood of zero. Consider W′:=W∩Tα−1W. Then, W′ is a nonempty neighborhood of zero. By hypothesis, t>0 can be selected such that(19)TtU∩W′≠ϕ,Tt+αW′∩U≠ϕ.
Hence,(20)TtU∩W∩Tα−1W≠ϕ,Tt+αW∩Tα−1W∩U≠ϕ.
Therefore,(21)TtU∩Tα−1W≠ϕ,Tt+αW∩U≠ϕ.
So,(22)Tt+αU∩W≠ϕ,Tt+αW∩U≠ϕ.
Now, by Theorem6, Ttt≥0 is recurrent.
In Theorem7, α is a positive and arbitrary scalar. So, by considering α=1, we get the following corollary.Corollary 4.
Suppose thatTtt≥0 is a C0-semigroup on X. If for any open and nonempty set U and any neighborhood W of zero, t>0 can be found so that(23)TtU∩W≠ϕ,Tt+1W∩U≠ϕ,then Ttt≥0 is recurrent.Proof.
It is enough to considerα=1 in Theorem 7
Like Theorem 4.1 in [14], we can define another recurrent criterion that is based on dense subsets.Theorem 8.
Assume thatTtt≥0 be a C0-semigroup on X. Suppose that there is a dense subset Y of X and there is a sequence tn of positive real numbers such that(i)
For anyy∈Y, Ttny⟶0(ii)
For anyy∈Y, there is yn in X such that yn⟶0 and Ttnyn⟶y
Then,Ttt≥0 is recurrent.Proof.
Assume thatU be an open and nonempty set. By density of Y, there is y∈U∩Y. Hence, ϵ>0 can be chosen such that By,ϵ⊆U.
By (i),Ttny⟶0, and by (ii), yn can be found such that yn⟶0 and Ttnyn⟶y. So, for sufficiently large N, we have(24)Ttny<ϵ3,yn<ϵ3,Ttnyn−y<ϵ3.
Therefore, if we considerx:=y+yn, then by (24),(25)x−y<ϵ,Ttnx−y<ϵ.
Hence,x∈U and Ttnx∈U. It means that Ttn−1U∩U≠ϕ. Thus, Ttt≥0 is recurrent.
By Theorem 2.3 in [15], the idea of the next theorem comes to mind.Theorem 9.
AssumeTtt≥0 be a C0-semigroup on X. Suppose that a dense subset Y of X exists such that(i)
For anyy∈Y, Tty⟶0 when t⟶0(ii)
For anyy∈Y and for any ϵ>0, x∈X and t>0 exist such that(26)x<ϵ,Ttx−y<ϵ.
Then,Ttt≥0 is recurrent.Proof.
Suppose thatU be an open and nonempty set. Let y∈U∩Y, and let ϵ>0 be such that By,ϵ⊆U. By (i), there is t0>0 such that(27)Tty<ϵ3,for anyt≥t0.
By (ii), there isx∈X such that(28)x<ϵ,Ttx−y<ϵ3.
So, if we considerz:=x+y, by (27) and (28), similar to the proof of Theorem 8, t>0 can be found such that z∈Tt−1U∩U.
It is considerable that if aC0-semigroup fulfills the conditions of Theorem 9, then it fulfills the conditions of Theorem 8.
The next sufficient condition for recurrence is based on special eigenvectors of operators of aC0-semigroup.Theorem 10.
ConsumeTtt≥0 is a C0-semigroup on X. Assume that λ<1 and tn⊆ℝ+ exist such that(29)Y=spanz∈X:Ttnz=λnz,n∈ℕ,be dense in X. Then, Ttt≥0 is recurrent.Proof.
Assume thatU is an open and nonempty subset of X. By density of Y, we can find x∈U∩Y. So, we can write x=∑i=1pαizi, where Ttnzi=λnzi and αi∈ℂ for 1≤i≤p. Now,(30)Ttnx=∑i=1pαiTtnzi=∑i=1pαiλnzi⟶0.
If we consideryn=∑i=1pαi/λnzi, then yn⟶0 and Ttnyn⟶x since(31)Ttnyn=∑i=1pαiλnTtnzi=∑i=1pαizi=x.
Hence,x+yn⟶x and(32)Ttnx+yn=Ttnx+Ttnyn⟶x.
So, for a large enoughN, x+yN∈U and TtNx+yn∈U. Accordingly, Ttt≥0 is recurrent.
By Theorem10, we can state the following interesting corollary.Corollary 5.
ConsumeTtt≥0 is a C0-semigroup on X. Assume that λ<1 is a scalar. If there is s>0 such that(33)Y=spanz∈X:Tsz=λz,be dense in X, then Ttt≥0 is recurrent.Proof.
Suppose thaty∈Y. Hence, y=∑i=1mαizi, where Tsnzi=λnzi. But Tsn=Tns. So, if we consider tn:=ns, by Theorem 10, Ttt≥0 is recurrent.


## 5. Direct Sum ofC0-Semigroups

We investigate the direct sum of recurrentC0-semigroups and their properties in this section. We begin this section by showing that recurrence of the direct sum of two C0-semigroups implies recurrence of each of them.Theorem 11.
IfSt⊕Ttt≥0 be a recurrent C0-semigroup on X⊕Y, then Stt≥0 and Ttt≥0 are recurrent on X and Y, respectively.Proof.
Assume thatU be an open and nonempty set in X. Then, U⊕0 is an open set of X⊕Y. By hypothesis, St⊕Ttt≥0 is recurrent. So, there exists t>0 so that(34)St⊕Tt−1U⊕0∩U⊕0≠ϕ.
Hence,St−1U∩U≠ϕ. This indicates that Stt≥0 is recurrent. Similarly, Ttt≥0 is recurrent.
It is proved that the recurrence ofSt⊕Ttt≥0 implies the recurrence of Stt≥0 and Ttt≥0. But if the converse is true?
In the following theorem, we state that the answer to the question is positive if at least one of them be mixing.Theorem 12.
Suppose thatStt≥0 is a recurrent C0-semigroup on X, and suppose that Ttt≥0 is a C0-semigroup on Y.(i)
IfTtt≥0 be a mixing semigroup, then St⊕Ttt≥0 is recurrent(ii)
If there ist0>0 such that Tnt0n∈ℕ be mixing, then St⊕Ttt≥0 is recurrentProof.
To prove part (i), consumeU⊕V is an open subset of X⊕Y. By hypothesis, Ttt≥0 is mixing. So, there is t1>0 such that, for any t≥t1,(35)Tt−1V∩V≠ϕ.
Also,Stt≥0 is recurrent. So, there is t2>0 such that(36)St2−1V∩V≠ϕ.
By Lemma1, we can assume that t2>t1. Hence, by (35) and (36),(37)Tt2⊕St2−1U⊕V∩U⊕V=Tt2−1U⊕St2−1V∩U⊕V=Tt2−1U∩U⊕St2−1V∩V≠ϕ.
For proving part (ii), note that, by Proposition 7.21 in [4], mixing of Tnt0n∈ℕ implies that Ttt≥0 is mixing. Hence, St⊕Ttt≥0 is recurrent by part (i).
The following corollary can be deduced from Theorem12.Corollary 6.
ConsumeStt≥0 and Ttt≥0 are mixing C0-semigroups on X and Y, respectively. Then, St⊕Ttt≥0 is recurrent.Especially,St⊕Stt≥0 and Tt⊕Ttt≥0 are recurrent.So, notable examples can be made by using mixingC0-semigroups as follows.Example 3.
Considerenn∈ℕ to be a sequence in X with this property that spanen:n∈ℕ¯=X. Consider B is an operator on X so that Be1=0 and Ben=en−1. It is proved that the generated C0-semigroup by B is mixing (Theorem 1.6 in [16]). So, if we denote this C0-semigroup with Ttt≥0, then by Corollary 6, Ttt≥0 is recurrent.
Also, we can state the following lemmas by using RHCC and HCC.Lemma 3.
IfStt≥0 is RHCC and Ttt≥0 is HCC, then St⊕Ttt≥0 is recurrent.Proof.
It is deduced from Lemma 5.3 in [8] that St⊕Ttt≥0 is HCC. Hence, it is recurrent.Lemma 4.
IfTtt≥0 is HCC, then Tt⊕Ttt≥0 is recurrent. Especially, Ttt≥0 is recurrent.Proof.
IfTtt≥0 is HCC, then by Theorem 2.5 in [16] and Theorem 7.28 in [4], Tt⊕Ttt≥0 is hypercyclic, and so, it is recurrent.
Also, we make the following example by Lemma4.Example 4.
LetI=0,∞, and let ρ denote an admissible weight function on I. Assume that X=LpρI,ℝ. If inft⟶∞ρt=0, then Ttt≥0 fulfills the hypercyclicity criterion (Proposition 4.4 in [8]). Then, Tt⊕Ttt≥0 is recurrent.



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