Codiskcyclic sets of operators on complex topological vector spaces
aa r X i v : . [ m a t h . F A ] F e b CODISKCYCLIC SETS OF OPERATORS ON COMPLEX TOPOLOGICALVECTOR SPACES
MOHAMED AMOUCH AND OTMANE BENCHIHEB
Abstract.
Let X be a complex topological vector space and L ( X ) the set of all continuouslinear operators on X. In this paper, we extend the notion of the codiskcyclicity of a singleoperator T ∈ L ( X ) to a set of operators Γ ⊂ L ( X ) . We prove some results for codiskcyclicsets of operators and we establish a codiskcyclicity criterion. As an application, we studythe codiskcyclicity of C -semigroups of operators. Introduction and Preliminary
Let X be a complex topological vector space and L ( X ) the set of all continuous linearoperators on X. By an operator, we always mean a continuous linear operator.The most studied notion in the linear dynamics is that of hypercyclicity : an operator T acting on X is called hypercyclic if there exits a vector x ∈ X such that the orbit of x under T ; Orb ( T, x ) = { T n x : n ∈ N } , is dense in X , such a vector x is called hypercyclic for T . The set of all hypercyclic vectors for T is denoted by HC ( T ) . Another important notion in the linear dynamics is that of supercyclicitywhich was introduced in [12]. We say that T is supercyclic if there exists x ∈ X such that C Orb ( T, x ) = { αT n x : α ∈ C , n ∈ N } , is dense in X . The vector x is called a supercyclic vector for T . We denote by SC ( T ) the setof all supercyclic vectors. For more information about hypercyclic and supercyclic operators,see [7, 11, 10].Another notion in the linear dynamics which was been studied by many authors is that ofcodiscyclicity : an operator T is called codiskcyclic if there is x ∈ X such that the codisk orbitof x under T ; U Orb ( T, x ) = { αT n x : α ∈ U , n ≥ } , is dense in X , when U := { α ∈ C : | α | ≥ } . In this case, the vector x is called a codiskcyclicvector for T . The set of all codiskcyclic vectors for T is denoted by U C ( T ) . In the case ofa separable complex Banach space, an operator T is codiskcyclic if and only if it is codisktransitive; that is for each pair ( U, V ) of nonempty open sets there exist some α ∈ U andsome n ≥ such that αT n ( U ) ∩ V = ∅ . For a general overview of the codiskcyclicity, see[13, 14, 17, 19].Recently, some notions of the linear dynamical system were introduced for a set Γ of operatorsinstead of a single operator T, see [1, 2, 3, 4, 5, 6] : A set Γ of operators is called hypercyclic ifthere exists a vector x in X such that its orbit under Γ ; Orb (Γ , x ) = { T x : T ∈ Γ } , is a densesubset of X . If there exits a vector x ∈ X such that C Orb (Γ , x ) = { αT x : T ∈ Γ , α ∈ C } , isa dense subset of X , then Γ is supercyclic. If x ∈ X is a vector such that span { Orb (Γ , x ) } = span { T x : T ∈ Γ } is dense in X , then Γ is cyclic. If there exists a vector x ∈ X such that itsdisk orbit under T ; D Orb (Γ , x ) = { αT x : T ∈ Γ , α ∈ D } , is a dense subset X , then Γ is calleda diskcyclic, when D is the unit closed disk. In each case, the vector x is called a hypercyclic,a supercyclic, a cyclic and a diskcyclic vector for Γ , respectively. Mathematics Subject Classification.
Key words and phrases.
Hypercyclity, supercyclicity, diskcyclicity, codiskcyclicity, C -semigroup. In this paper, we continue the study of the dynamics of a set of operator by introducing theconcept of codiskcyclicity for a set of operators.In Section , we introduce and study the codiskcyclicity for a set of operators. In partic-ular, we show that the set of codiscyclic vectors of a set Γ is a G δ type and we prove thatcodiskcyclicity is preserved under quasi-similarity.In Section , we extend the notion of codisk transitivity of a single operator to a set ofoperators. We give the relation between this notion and the concept of codiskcyclic and weestablish a codiskcyclic criterion.In Section , we study the codiskcyclicity of a C -semigroup of operators. We show thatthe codiskcyclicity and the codisk transitivity are equivalent and we prove that a codiskcyclic C -semigroup of operators exists on X if and only if dim ( X ) = 1 or dim ( X ) = ∞ .2. Codiskcyclic Sets of Operators
In the following definition, we introduce the notion of the codiskcyclicity of a set of operatorsinstead of a single operator.
Definition 2.1.
We say that Γ is codiskcyclic if there exists x ∈ X for which the codisk orbitof x under Γ; U Orb (Γ , x ) := { αT x : α ∈ U , T ∈ Γ } , is dense in X . The vector x is called a codiskcyclic vector for Γ . The set of all codiskcyclicvectors for Γ is denoted by U C (Γ) . Remark . An operator T is codiskcyclic if and only if the set Γ = { T n : n ≥ } is codiskcyclic. Example 2.3.
Let f be a nonzero linear form on a locally convex space X and D be a subsetof X such that the set U D := { αx : α ∈ U , x ∈ D } is a dense subset of X . For all x ∈ X , let T x defined by T x y = f ( y ) x, for all y ∈ X. Put Γ f = { T x : x ∈ D } and let y be a vector of X suchthat f ( y ) = 0 . Then U Orb (Γ f , y ) = { αT x y : x ∈ D , α ∈ U } = { αf ( y ) x : x ∈ D , α ∈ U } = U D. Hence, Γ f is codiskcyclic.A necessary condition for the codiskcyclicity is given by the following proposition. Proposition 2.4.
Let X be a complex normed space and Γ a subset of L ( X ) . If x is acodiskcyclic vector for Γ) , then sup {k αT x k : α ∈ U , T ∈ Γ } = + ∞ . Proof.
Let x ∈ U C (Γ) . Assume that sup {k αT x k : α ∈ U , T ∈ Γ } = m < + ∞ , and let y ∈ X such that k y k > m . Since x ∈ U C (Γ) , there exist { α k } ⊂ U and { T k } ⊂ Γ such that α k T k x −→ y. Hence, k y k ≤ m , which is a contradiction. (cid:3) We denote by { Γ } ′ the set of all elements of L ( X ) which commutes with every element of Γ . Proposition 2.5.
Let X be a complex topological vector space and Γ a subset of L ( X ) . Assumethat Γ is codiskcyclic and let T ∈ L ( X ) be with dense range. If T ∈ { Γ } ′ , then T x ∈ U C (Γ) ,for all x ∈ U C (Γ) .Proof. Let O be a nonempty open subset of X . Since T is continuous and of dense range, T − ( O ) is a nonempty open subset of X . Let x ∈ U C (Γ) , then there exist α ∈ U and S ∈ Γ such that αSx ∈ T − ( O ) , that is αT ( Sx ) ∈ O . Since T ∈ { Γ } ′ , it follows that αS ( T x ) = αT ( Sx ) ∈ O. Hence, U Orb (Γ , T x ) meets every nonempty open subset of X . From this, U Orb (Γ , T x ) is densein X . That is, T x ∈ U C (Γ) . (cid:3) Recall from [2], that Γ ⊂ L ( X ) and Γ ⊂ L ( Y ) are called quasi-similar if there exists acontinuous map φ : X −→ Y with dense range such that for all T ∈ Γ , there exists S ∈ Γ satisfying S ◦ φ = φ ◦ T . If φ can be chosen to be a homeomorphism, then Γ and Γ are calledsimilar.In the following, we prove that the codiskcyclicity is preserved under quasi-similarity. Proposition 2.6. If Γ and Γ are quasi-similar, then Γ is codiskcyclic in X implies that Γ iscodiskcyclic in Y . Moreover, φ ( U C (Γ) ⊂ U C (Γ ) . ODISKCYCLIC SETS OF OPERATORS ON COMPLEX TOPOLOGICAL VECTOR SPACES 3
Proof.
Assume that Γ is codiskcyclic in X . Let O be a nonempty open subset of Y , then φ − ( O ) is a nonempty open subset of X . If x ∈ U C (Γ) , then there exist α ∈ U and T ∈ Γ such that αT x ∈ φ − ( O ) , that is αφ ( T x ) ∈ O . Let S ∈ Γ such that S ◦ φ = φ ◦ T . Hence, αS ( φx ) = αφ ( T x ) ∈ O. Hence, Γ is codiskcyclic and φx ∈ U C (Γ ) . (cid:3) Proposition 2.7.
Let ( c T ) T ∈ Γ ⊂ R ∗ + . If { c T T : T ∈ Γ } is codiskcyclic and ( k T ) T ∈ Γ is suchthat c T ≥ k T > for all T ∈ Γ , then the set { k T T : T ∈ Γ } is codiskcyclic.Proof. Let x be a codiskcyclic vector for { c T T : T ∈ Γ } . Since c T ≥ k T for all T ∈ Γ , we have U Orb ( { c T T : T ∈ Γ } , x ) ⊂ U Orb ( { k T T : T ∈ Γ } , x ) . Since U Orb ( { c T T : T ∈ Γ } , x ) is densein X , it follows that U Orb ( { k T T : T ∈ Γ } , x ) is dense in X , this means that { k T T : T ∈ Γ } iscodiskcyclic in X . (cid:3) Proposition 2.8.
Let { X i } ni =1 be a family of complex topological vector spaces and Γ i a subsetof L ( X i ) , for ≤ i ≤ n . If L ni =1 Γ i is a codiskcyclic set in L ni =1 X i , then Γ i is a codiskcyclicset in X i , for all ≤ i ≤ n .Proof. If ≤ j ≤ n , then L ni =1 Γ i is quasi-similar to Γ j , and the result follow by Proposition2.6. (cid:3) Let X be a complex topological vector space. The following proposition gives a characteriza-tion of the set of codiskcyclic vector of set of operators using a countable basis of the topologyof X . Note that the set D is the unit closed disk defined by D = { α ∈ C : | α | ≤ } . Proposition 2.9.
Let X be a second countable complex topological vector space and Γ a subsetof L ( X ) . If Γ is codiskcyclic, then U C (Γ) = \ n ≥ [ β ∈ D [ T ∈ Γ T − ( βU n ) , where ( U n ) n ≥ is a countable basis of the topology of X . As a consequence, U C (Γ) is a G δ typeset.Proof. Let x ∈ X . Then, x ∈ U C (Γ) if and only if U Orb (Γ , x ) = X . Equivalently, for all n ≥ , U n ∩ U Orb (Γ , x ) = ∅ , that is for all n ≥ , there exist λ ∈ U and T ∈ Γ such that λT x ∈ U n . This is equivalent to the fact that for all n ≥ , there exist β ∈ D and T ∈ Γ suchthat x ∈ T − ( βU n ) . Hence, x ∈ \ n ≥ [ β ∈ D [ T ∈ Γ T − ( βU n ) . (cid:3) Density and Codisk Transitivity of Sets of Operators
In the following definition, we introduce the notion of a codisk transitive set of operatorswhich generalize the codisk transitivity of a single operator.
Definition 3.1.
We say that Γ is a codisk transitive set of operators, if for any pair ( U, V ) ofnonempty open subsets of X , there exist α ∈ U and T ∈ Γ such that T ( αU ) ∩ V = ∅ . Remark . An operator T ∈ L ( X ) is codisk transitive if and only if Γ = { T n : n ≥ } iscodisk transitive. Example 3.3.
Assume that X is a locally convex space. Let x , y ∈ X and let f y be a linearform on X such that f y ( y ) = 0 . Let T f y ,x be an operator defined by T f y ,x z = f y ( z ) x. Define
Γ = { T f y ,x : x , y ∈ X such that f y ( y ) = 0 } . Let U and V be two nonempty open subsets of X . There exist x, y ∈ X such that x ∈ U and y ∈ V . We have T f y ,x ( y ) = f y ( y ) x. Since < f y ( y ) < , it follows that x = f y ( y ) T f y ,x ( y ) . Hence x ∈ U and x ∈ f y ( y ) T f y ,x ( V ) , whichimplies that U ∩ f y ( y ) T f y ,x ( V ) = ∅ . Thus Γ is a codisk transitive.In the following proposition, we prove that the codisk transitivity of sets of operators ispreserved under quasi-similarity. MOHAMED AMOUCH AND OTMANE BENCHIHEB
Proposition 3.4.
Assume that Γ ⊂ L ( X ) and Γ ⊂ L ( Y ) are quasi-similar. If Γ is codisktransitive in X , then Γ is codisk transitive in Y .Proof. Let U and V be nonempty open subsets of X . Since φ is continuous and of dense range, φ − ( U ) and φ − ( V ) are nonempty and open sets. Since Γ is codisk transitive in X , thereexist y ∈ φ − ( U ) and α ∈ U , T ∈ Γ with αT y ∈ φ − ( V ) , which implies that φ ( y ) ∈ U and αφ ( T y ) ∈ V . Let S ∈ Γ such that S ◦ φ = φ ◦ T . Then, φ ( y ) ∈ U and αSφ ( y ) ∈ V . Thus, αS ( U ) ∩ V = ∅ . Hence, Γ is codisk transitive in Y. (cid:3) In the following result, we give necessary and sufficient conditions for a set of operators tobe codisk transitive.
Theorem 3.5.
Let X be a complex normed space and Γ a subset of L ( X ) . The followingassertions are equivalent :( i ) Γ is codisk transitive; ( ii ) For each x , y ∈ X, there exists sequences { x k } in X , { α k } in U and { T k } in Γ suchthat x k −→ x and α k T k ( x k ) −→ y ;( iii ) For each x , y ∈ X and for W a neighborhood of , there exist z ∈ X , α ∈ U and T ∈ Γ such that x − z ∈ W and αT ( z ) − y ∈ W. Proof. ( i ) ⇒ ( ii ) Let x , y ∈ X . For all k ≥ , let U k = B ( x, k ) and V k = B ( y, k ) . Then U k and V k are nonempty open subsets of X . Since Γ is codisk transitive, there exist α k ∈ U and T k ∈ Γ such that α k T k ( U k ) ∩ V k = ∅ . For all k ≥ , let x k ∈ U k such that α k T k ( x k ) ∈ V k , then k x k − x k < k and k α k T k ( x k ) − y k < k , this implies that x k −→ x and α k T k ( x k ) −→ y. ( ii ) ⇒ ( iii ) Clear. ( iii ) ⇒ ( i ) Let U and V be two nonempty open subsets of X . Then there exists x , y ∈ X suchthat x ∈ U and y ∈ V . Since for all k ≥ , W k = B (0 , k ) is a neighborhood of , there exist z k ∈ X, α k ∈ U and T k ∈ Γ such that k x − z k k < k and α k T k ( z k ) − y k < k . This implies that z k −→ x and α k T k ( z k ) −→ y. Since U and V are nonempty open subsets of X , x ∈ U and y ∈ V , there exists N ∈ N such that z k ∈ U and α k T k ( z k ) ∈ V , for all k ≥ N. (cid:3) Let Γ be a subset of L ( X ) . In whats follows, we prove that Γ is codisk transitive if and onlyif it admits a dense subset of codiskcyclic vectors. Theorem 3.6.
Let X be a second countable Baire complex topological vector space and Γ asubset of L ( X ) . The following assertions are equivalent :( i ) U C (Γ) is dense in X ; ( ii ) Γ is codisk transitive.As a consequence, a codisk transitive set is codiskcyclic.Proof. ( i ) ⇒ ( ii ) : Assume that U C (Γ) is dense in X and let U and V be two nonempty opensubsets of X . By Proposition 2.9, we have U C (Γ) = T n ≥ (cid:16)S β ∈ D S T ∈ Γ T − ( βU n ) (cid:17) . Hence, forall n ≥ , A n := [ β ∈ D [ T ∈ Γ T − ( βU n ) is dense in X . Thus, for all n, m ≥ , we have A n ∩ U m = ∅ which implies that for all n, m ≥ , there exist β ∈ U and T ∈ Γ such that T ( βU m ) ∩ U n = ∅ .Hence, Γ is a codisk transitive set. ( ii ) ⇒ ( i ) : Assume that Γ is codisk transitive. Let n , m ≥ , then there exist β ∈ U and T ∈ Γ such that T ( βU m ) ∩ U n = ∅ , which implies that T − ( β U n ) ∩ U m = ∅ . Hence, for all n ≥ , we have [ β ∈ D [ T ∈ Γ T − ( βU n ) is dense in X . Since X is a Baire space, it follows that U C (Γ) = T n ≥ (cid:16)S β ∈ D S T ∈ Γ T − ( βU n ) (cid:17) is a dense subset of X . (cid:3) The converse of Theorem 3.6 holds with some additional assumption.
ODISKCYCLIC SETS OF OPERATORS ON COMPLEX TOPOLOGICAL VECTOR SPACES 5
Theorem 3.7.
Let X be a complex topological vector space and Γ a subset of L ( X ) . Assumethat for all T , S ∈ Γ with T = S , there exists A ∈ Γ such that T = AS . The following assertionsare equivalent :( i ) Γ is codiskcyclic; ( ii ) Γ is codisk transitive.Proof. ( i ) ⇒ ( ii ) This implication is due to Theorem 3.6. ( ii ) ⇒ ( i ) Since Γ is codiskcyclic, there exists x ∈ X such that U Orb (Γ , x ) is a dense subset of X . Let U and V be two nonempty open subsets of X , then there exist α , β ∈ U with | α | ≥ | β | ,and T , S ∈ Γ such that αT x ∈ U and βSx ∈ V. There exists A ∈ Γ such that T = AS . Hence, αA ( Sx ) ∈ U and βA ( Sx ) ∈ A ( V ) , which implies that U ∩ A ( αβ V ) = ∅ . Hence, Γ is codisktransitive. (cid:3) In the following definition we introduce the notion of strictly codisk transitivity of a set ofoperators. The case of hypercyclicity (resp, supercyclicity, diskcyclicity) were introduced in[1, 4, 5].
Definition 3.8.
We say that Γ is strictly codisk transitive if for each pair of nonzero elements x, y in X , there exist some α ∈ U and T ∈ Γ such that αT x = y. Remark . An operator T ∈ L ( X ) is strictly codisk transitive if and only if the set Γ = { T n : n ≥ } is a strictly codisk transitive. Proposition 3.10. If Γ is strictly codisk transitive set, then it is codisk transitive. As aconsequence, if Γ is strictly codisk transitive set, then it is codiskcyclic.Proof. Assume that Γ is a strictly codisk transitive set. If U and V are two nonempty opensubsets of X , then there exist x, y ∈ X such that x ∈ U and y ∈ V . Since Γ is strictly codisktransitive, it follows that there exist α ∈ U and T ∈ Γ such that αT x = y. Hence, αT x ∈ αT ( U ) and αT x ∈ V. Thus, αT ( U ) ∩ V = ∅ , which implies that Γ is codisk transitive. By Theorem3.6, we deduce that Γ is codiskcyclic. (cid:3) In the following proposition, we prove that the strictly codisk transitivity of sets of operatorsis preserved under similarity.
Proposition 3.11. If Γ ⊂ L ( X ) and Γ L ( Y ) are similar, then Γ is strictly codisk transitive in X if and only if Γ is strictly codisk transitive in Y. Proof.
Let x , y ∈ Y . There exist a, b ∈ X such that φ ( a ) = x and φ ( b ) = y . Since Γ is strictlycodisk transitive in X , there exist α ∈ U and T ∈ Γ such that αT a = b, this implies that αφ ◦ T ( a ) = φ ( b ) . Let S ∈ Γ such that S ◦ φ = φ ◦ T . Hence, αSx = y . Hence Γ is strictlycodisk transitive in Y . (cid:3) Recall that the strong operator topology (SOT for short) on L ( X ) is the topology withrespect to which any T ∈ L ( X ) has a neighborhood basis consisting of sets of the form Ω = { S ∈ L ( X ) : Se i − T e i ∈ U , i = 1 , , . . . , k } , where k ∈ N , e , e , . . . e k ∈ X are linearly independent and U is a neighborhood of zero in X ,see [8].Let x be an element of a complex topological vector space X . Note that U x is the subset of X defined by U { x } := U x = { αx : α ∈ U } . In the following theorem, the proof is also true for norm-density if X is assumed to be anormed linear space. Theorem 3.12.
For each pair of nonzero vectors x , y ∈ X with y / ∈ U x , there exists a SOT-dense set Γ xy ⊂ L ( X ) which is not strictly codisk transitive. Furthermore, Γ ⊂ L ( X ) is a densenonstrictly codisk transitive set if and only if Γ is a dense subset of Γ xy for some x , y ∈ X. MOHAMED AMOUCH AND OTMANE BENCHIHEB
Proof.
Fix nonzero vectors x, y ∈ X such that y / ∈ U x and let Γ xy the set defined by Γ xy = { T ∈ L ( X ) : y / ∈ U T x } . Then Γ xy is not strictly codisk transitive. Let Ω be a nonempty open set in L ( X ) and S ∈ Ω .If Sx and y are such that y / ∈ U Sx , then S ∈ Ω ∩ Γ xy . Otherwise, putting S n = S + n I , we seethat S k ∈ Ω for some k , but S k x and y are such that y / ∈ U S k x . Hence, Ω ∩ Γ xy = ∅ and theproof is completed.We prove the second assertion of the theorem. Suppose that Γ is a dense subset of L ( X ) thatis not strictly codisk transitive. Then there are nonzero vectors x, y ∈ X such that y / ∈ U T x for all T ∈ Γ and hence Γ ⊂ Γ xy . To show that Γ is dense in Γ xy , assume that Ω is an opensubset of Γ xy . Thus, Ω = Γ xy ∩ Ω for some open set Ω in L ( X ) . Then Γ ∩ Ω = Γ ∩ Ω = ∅ .For the converse, let Γ be a dense subset of Γ xy for some x, y ∈ X . Then Γ is not strictlycodisk transitive. Also, since Γ xy is a dense open subset of L ( X ) , we conclude that Γ is alsodense in L ( X ) . Indeed, if Ω is any open set in L ( X ) then Ω ∩ Γ xy = ∅ since Γ xy is dense in L ( X ) . On the other hand, Ω ∩ Γ xy is open in Γ xy and so it must intersect Γ since Γ is dense in Γ xy . Thus, Ω ∩ Γ = ∅ and so Γ is dense in L ( X ) . (cid:3) Corollary 3.13.
There is a subset Γ of Γ such that Γ = L ( X ) and Γ is not strictly codisktransitive.Proof. For nonzero x, y such that y / ∈ U x , put Γ = Γ ∩ Γ xy . (cid:3) In the following definition, we introduce that notion of codiskcyclic transitivity of set ofoperators. The case of hypercyclicity (resp, supercyclicity, diskcyclicity) were introduced in[1, 4, 5].
Definition 3.14.
We say that Γ is a codiskcyclic transitive set or codiskcyclic transitive if U C (Γ) = X \ { } . Remark . An operator T ∈ L ( X ) is codiskcyclic transitive if and only if the set Γ = { T n : n ≥ } is codiskcyclic transitive.It is clear that a codiskcyclic transitive set is codiskcyclic. Moreover, the next propositionshows that codiskcyclic transitivity of sets of operators implies codisk transitivity. Proposition 3.16. If Γ is codiskcyclic transitive, then Γ is codisk transitive.Proof. Let U and V be two nonempty open subsets of X . There exists x ∈ X \ { } such that x ∈ U . Since Γ is codiskcyclic transitive, there exists α ∈ U and T ∈ Γ such that αT x ∈ V .This implies that αT ( U ) ∩ V = ∅ . Hence, Γ is codisk transitive. (cid:3) In the following proposition, we prove that the codiskcyclic transitivity is preserved undersimilarity.
Proposition 3.17.
Assume that Γ and Γ are similar, then Γ is codiskcyclic transitive on X if and only if Γ is codiskcyclic transitive on Y .Proof. If Γ is a codiskcyclic transitive on X , then by Proposition 2.6, φ ( U C (Γ)) ⊂ U C (Γ ) .Since φ is homeomorphism, the result holds. (cid:3) Assume that X is a topological vector space and Γ a subset of L ( X ) . The following resultshows that the SOT-closure of Γ is not large enough to have more codiskcyclic vectors than Γ . Proposition 3.18. If Γ stands for the SOT-closure of Γ then U C (Γ) = U C (Γ) . Proof.
We only need to prove that U C (Γ) ⊂ U C (Γ) . Fix x ∈ U C (Γ) and let U be an arbitraryopen subset of X . Then there is some α ∈ U and T ∈ Γ such that αT x ∈ U . The set Ω = { S ∈ L ( X ) : αSx ∈ U } is a SOT-neighborhood of T and so it must intersect Γ . Therefore,there is some S ∈ Γ such that αSx ∈ U and this shows that x ∈ U C (Γ) . (cid:3) ODISKCYCLIC SETS OF OPERATORS ON COMPLEX TOPOLOGICAL VECTOR SPACES 7
Corollary 3.19.
Let X be a topological vector space and Γ a subset of L ( X ) . Then Γ iscodiskcyclic transitive if and only if Γ is codiskcyclic transitive.Proof. Assume that Γ is codiskcyclic transitive, then U C (Γ) = X \ { } . Since by Proposition3.18, we have U C (Γ) = U C (Γ) , it follows that U C (Γ) = X \ { } . Hence, Γ is codiskcyclictransitive. (cid:3) In the next definition, we introduce the notion of codiskcyclic criterion of a set of operatorswhich generalizes the definition of codiskcyclic criterion of a single operator.
Definition 3.20.
We say that Γ satisfies the criterion of codiskcyclicity if there exist two densesubsets X and Y in X and sequences { α k } of U , { T k } of Γ and a sequence of maps S k : Y −→ X such that :( i ) α k T k x −→ for all x ∈ X ; ( ii ) α − k S k x −→ for all y ∈ Y ; ( iii ) T k S k y −→ y for all y ∈ Y . Remark . An operator T ∈ L ( X ) satisfies the criterion of codiskcyclicity for operators if andonly if the set Γ = { T n : n ≥ } satisfies the criterion of codiskcyclicity for sets of operators,see [19]. Theorem 3.22.
Let X be a second countable Baire complex topological vector space and Γ asubset of L ( X ) . If Γ satisfies the criterion of codiskcyclicity, then U C (Γ) is a dense subset of X . As consequence; Γ is codiskcyclic.Proof. Let U and V be two nonempty open subsets of X . Since X and Y are dense in X , there exist x and y in X such that x ∈ X ∩ U and y ∈ Y ∩ V. For all k ≥ , let z k = x + α − k S k y . We have α − k S k y −→ , which implies that z k −→ x . Since x ∈ U and U is open, there exists N ∈ N such that z k ∈ U , for all k ≥ N . On the other hand, we have α k T k z k = α k T k x + T k ( S k y ) −→ y . Since y ∈ V and V is open, there exists N ∈ N suchthat α k T k z k ∈ V , for all k ≥ N . Let N = max { N , N } , then z k ∈ U and α k T k z k ∈ V , for all k ≥ N , that is α k T k ( U ) ∩ V = ∅ , for all k ≥ N . Hence, Γ is codisk transitive. By Theorem 3.6we deduce that U C (Γ) is a dense subset of X . We use again Theorem 3.6 to conclude that Γ is codiskcyclic and this complete the proof. (cid:3) Codiskcyclic C -Semigroups of Operators In this section we will study the particular case when Γ is a C -semigroup of operators.Recall that a family ( T t ) t ≥ of operators is called a C -semigroup of operators if the followingthree conditions are satisfied :( i ) T = I the identity operator on X ; ( ii ) T t + s = T t T s for all t, s ≥ ; ( iii ) lim t → s T t x = T s x for all x ∈ X and t ≥ .For more informations about the theory of C -semigroups the reader may refer to [16]. Example 4.1.
Let X = C . For all t ≥ , let T t x = exp( t ) x , for all x ∈ C . Then ( T t ) t ≥ is a C -semigroup and we have U Orb (( T t ) t ≥ ,
1) = { αT t (1) : t ≥ , α ∈ U } = { αy : y ∈ R + , α ∈ U } . Let x ∈ C \ { } . Then x = | x | x | x | ∈ U Orb (( T t ) t ≥ , . Hence, U Orb (( T t ) t ≥ ,
1) = C . Thus, ( T t ) t ≥ is a codiskcyclic C -semigroup of operators and is a codiskcyclic vector for ( T t ) t ≥ .Recall from [18, Lemma 5.1], that if X is a complex topological vector space such that ≤ dim ( X ) < ∞ , then X supports no supercyclic C -semigroups of operators.In the following theorem we will prove that the same result holds in the case of codiskcyclicityon a complex topological vector space. Theorem 4.2.
Assume that ≤ dim ( X ) < ∞ . Then X supports no codiskcyclic C -semigroups.Proof. By using [18, Lemma 5.1] and the fact that U Orb (Γ , x ) ⊂ C Orb (Γ , x ) . (cid:3) MOHAMED AMOUCH AND OTMANE BENCHIHEB
A necessary and sufficient condition for a C -semigroup of operators to be codiskcyclic isgiven in the next lemma and theorem. Lemma 4.3.
Let ( T t ) t ≥ be a codiskcyclic C -semigroup of operators on a Banach infinitedimensional space X . If x ∈ X is a codiskcyclic vector of ( T t ) t ≥ , then the following assertionshold: (1) T t x = 0 , for all t ≥ ; (2) The set { αT t x : t ≥ s , α ∈ U } is dense in X , for all s ≥ .Proof. (1) Suppose that t > is minimal with the property that T t x = 0 . We show first thateach y ∈ X is of the form y = αT t x for some t ∈ [0 , t ] and α ∈ U . Since x ∈ U C (Γ) , there exista sequence ( t n ) n ∈ N ⊂ [0 , t ] and a sequence ( α n ) n ∈ N ⊂ U such that α n T t n x −→ y . Withoutloss of generality we may assume that ( t n ) n ∈ N converges to some t . By compactness we mayassume that ( α n ) n ∈ N converges to some α and we infer that y = αT t x .Now take three vectors y i = α i T t i x ∈ X , spanning a two-dimensional subspace, such thateach pair y i , y j , i = j , is linearly independent. Assume that t > t > t . We have then y = c y + c y . Now we arrive at the contradiction = α T ( t + t − t ) x = T ( t − t ) y = c T ( t − t ) y + c T ( t − t ) y = c α T ( t + t − t ) x + c α T t x = 0 . (2) Suppose that there exists s > such that { αT t x : t ≥ s , α ∈ U } is not dense in X.Hence there exists a bounded open set U such that U ∩ A = ∅ . Therefore we have U ⊂{ αT t x : ≤ t ≤ s , α ∈ U } by using the relation X = { αT t x : t ≥ , α ∈ U } = { αT t x : t ≥ s , α ∈ U } ∪ { αT t x : ≤ t ≤ s , α ∈ U } . Thus, U is compact. Hence X is finite dimensional, which contradicts that X is infinite dimen-sional. (cid:3) Theorem 4.4.
Let ( T t ) t ≥ be a C -semigroup of operators on a separable Banach infinitedimensional space X. Then the following assertions are equivalent: (1) ( T t ) t ≥ is codiskcyclic; (2) for all y , z ∈ X and all ε > , there exist v ∈ X , t > and α ∈ U such that k y − v k < ε and k z − αT t v k < ε ;(3) for all y , z ∈ X , all ε > and for all l ≥ , there exist v ∈ X , t > l and α ∈ U suchthat k y − v k < ε and k z − αT t v k < ε. Proof. (1) ⇒ (3) : Let x ∈ X such that { αT t x : t ≥ , α ∈ U } is dense in X and let ε > . Forany y ∈ X , there exist s > and α ∈ U such that k y − α T s x k < ε . If l ≥ , then by Lemma4.3, the set α { αT t x : t ≥ s + l , α ∈ U } := { α αT t x : t ≥ s + l , α ∈ U } is dense in X . For any z ∈ X , there exist s > l + s and α ∈ U such that k z − α α T s x k < ε . Put v = α T s x , t = s − s > l and α = α . Then we have k y − v k < ε and k z − αT t v k < ε. (3) ⇒ (2) : It is obvious. (2) ⇒ (1) : Let { z , z , z , ... } be a dense sequence in X . we construct sequences { y , y , y , ... } ⊂ X , { t , t , t , ... } ⊂ [0 , + ∞ ) and { α , α , α , ... } ⊂ U inductively: • Put y = z , t = 0 ; • For n > , find y n , t n and α n such that k y n − y n − k ≤ − n sup {k T t j k : j < n } , (4.1)and k z n − α n T t n y n k ≤ − n . (4.2) ODISKCYCLIC SETS OF OPERATORS ON COMPLEX TOPOLOGICAL VECTOR SPACES 9
In particular, (4.1) implies that k y n − y n − k ≤ − n , so that the sequence ( y n ) n ≥ has a limit x . Applying (4.2) and once again (4.1) we infer that k z n − α n T t n x k = k z n − α n T t n y n + α n T t n y n − α n T t n x k≤ k z n − α n T t n y n k + k α n T t n ( y n − x ) k≤ k z n − α n T t n y n k + k α n T t n k + ∞ X i = n +1 k y i − y i − k≤ − n + + ∞ X i = n +1 − i = 2 − n +1 . Given z ∈ X and ε > there are arbitrarily large n such that k z n − z k < ε . Choosing n large enough such that − n +1 < ε , we obtain k α n T t n x − z k ≤ k z − z n k + k z n − α n T t n x k < ε. Therefore, { αT t x : t ≥ , α ∈ U } is dense in X . (cid:3) As a corollary we obtain a sufficient condition of codiskcyclicity of a C -semigroup of oper-ators.Let X be a separable Banach infinite dimensional space. Denote X the set of all x ∈ X such that lim t −→∞ T t x = 0 , and X ∞ the set of all x ∈ X such that for each ε > there existsome w ∈ X , α ∈ U and some t > with k w k < ε and k αT t w − x k < ε. Theorem 4.5.
Let ( T t ) t ≥ be a C -semigroup of operators on a separable Banach infinitedimensional space X . If both X ∞ and X are dense subsets, then ( T t ) t ≥ is codiskcyclic.Proof. Let z ∈ X ∞ and y ∈ X . Then for each ε > there are arbitrarily large t > , α ∈ U and w ∈ X such that k w k < ε and k αT t w − x k < ε . Since y ∈ X , for sufficiently large t wehave k αT t y k < ε . We put v = y + w and infer k z − T t v k ≤ k z − T t w k + k αT t y k < ε, and k y − v k = k w k < ε. By Theorem 4.4, the result holds. (cid:3)
We use Theorem 3.7 to prove that the codiskcyclicity and codisk transitivity of a C -semigroup of operators on a complex topological vector space are equivalent. Theorem 4.6.
Let ( T t ) t ≥ be a C -semigroup of operators on a complex topological vector space X . Then, the following assertions are equivalent :( i ) ( T t ) t ≥ is codiskcyclic; ( ii ) ( T t ) t ≥ is codisk transitive.Proof. By remarking that if t > t ≥ , then there exists t = t − t such that T t = T t T t ,and using Theorem 3.7. (cid:3) References [1] M. Amouch, O. Benchiheb, Diskcyclicity of sets of operators and applications. Acta Mathematica Sinica,English Series., (2020). 36, 1203-1220.[2] M. Amouch, O. Benchiheb, On cyclic sets of operators. Rend. Circ. Mat. Palermo, 2. Ser., (2019), 68,521-529.[3] M. Amouch, O. Benchiheb, On linear dynamics of sets of operators. Turk. J. Math., (2019), 43(1), 402-411.[4] M. Amouch, O. Benchiheb, Some versions of supercyclicity of a s set of operators. Accepted for publicationin Filomat Journal.[5] M. Ansari, K. Hedayatian, B. Khani-robati, On the density and transitivity of sets of operators, Turk. J.Math., 42 (2018), 181-189.[6] M. Ansari, K. Hedayatian, B. Khani-robati, A. Moradi, A note on topological and strict transitivity. Iran.J. Sci. Technol. Trans. Sci. 42(1), 59-64 (2018)[7] F. Bayart, E. Matheron, Dynamics of linear operators. New York, NY, USA: Cambridge University Press,(2009).[8] J.B. Conway, A course in Functional Analysis. Springer Graduate Texts in Math Series, 1985.[9] W. Desch, W. Schappacher, G. F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergod.Th. Dynam. Sys.,1997, 17, 793-819.[10] K.-G. Grosse-Erdmann, A. Peris Manguillot, Linear chaos, Universitext, Springer, London, 2011. [11] K. G. Grosse-Erdmann, (1999). Universal families and hypercyclic operators, Bulletin of the AmericanMathematical Society, 36(3), 345-381.[12] H. M. Hilden, L. J. Wallen, Some cyclic and non-cyclic vectors of certain operators, Indiana Univ. Math.J. 23 (1973/74), 557-565.[13] Y. X. Liang, Z. H. Zhou Disk-cyclic and Codisk-cyclic tuples of the adjoint weighted composition operatorson Hilbert spaces, Bull. Belg. Math. Soc. Simon Stevin., 23(2), 203-215 (2016).[14] Y. X. Liang, Z. H. Zhou, Disk-cyclic and Codisk-cyclic of certain shift operators, operators and matrices.,9(4), 831-846 (2015).[15] M. Matsui, M. Yamada. and F. Takeo, Erratum to Supercyclic and chaotic translation semigroups, Proc.Amer. Math. Soc., 2004, 132, 3751-3752.[16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.[17] Y. Wang, H. G. Zeng, Disk-cyclic and codisk-cyclic weighted pseudo-shifts, Bull. Belg. Math. Soc. SimonStevin., 25(2), 209-224 (2018).[18] J. Wengenroth, Hypercyclic operators on non-locally convex spaces, Proc. Amer. Math. Soc. 131 (2003)1759-1761.[19] Z. J. Zeana, Cyclic phenomena of operators on Hilbert space, Thesis, University of Baghdad (2002).
Mohamed Amouch and Otmane Benchiheb, University Chouaib Doukkali. Department of Math-ematics, Faculty of science Eljadida, Morocco
Email address : [email protected] Email address ::