Supercyclicity of Multiplication on Banach Ideal of Operators

abstract: Let X be a complex Banach space with dimX > 1 such that its topological dual X is separable and B(X) the algebra of all bounded linear operators on X. In this paper, we study the passage of property of being supercyclic from T ∈ B(X) to the left and the right multiplication induced by T on an admissible Banach ideal of B(X). Also, we give a sufficient conditions for the tensor product T ⊗̂R of two operators on B(X) to be supercyclic.


Introduction and Preliminary
Throughout this paper, let X be a Banach space with dim X > 1 such that its topological dual X * is separable, B(X) be the algebra of all bounded linear operators on X and K(X) be the algebra of all compact operators on X. For T ∈ B(X), the orbit of x ∈ X under T is the set Orb(T, x) := {T n x : n ∈ N}.
An operator T ∈ B(X) is said to be hypercyclic if there is some vector x ∈ X such that Orb(T, x) is dense in X; such a vector x is called a hypercyclic vector for T . Similarly, T ∈ B(X) is said to be supercyclic if there is some vector x ∈ X such that COrb(T, x) := {αT n x : α ∈ C, n ∈ N} is dense in X; such a vector x is called supercyclic vector for T . From [7], T ∈ B(X) is supercyclic if and only if for each pair (U, V ) of nonempty open subsets of X there exist α ∈ C and n ∈ N such that αT n (U ) ∩ V = ∅.
We say that an operator T ∈ B(X) satisfies the hypercyclicity criterion, if there exist two dense subsets D 1 , D 2 ⊂ X, a strictly increasing sequence (n k ) k≥0 of positive integers and a sequence of maps S n k : D 2 −→ X such that : (i) T n k x −→ 0 for any x ∈ D 1 ; (ii) S n k y −→ 0 for any y ∈ D 2 ; (iii) T n k S n k y −→ y for any y ∈ D 2 .
The hypercyclicity criterion was introduced in [15] and studied in [3,8,15]. In [21], Salas gave a characterization of the supercyclic bilateral backward weighted shifts in terms of the supercyclicity criterion that ensure supercyclicity. We say that T ∈ B(X) satisfies the supercyclicity criterion, if there exist two dense subsets D 1 and D 2 in X, a sequence (n k ) k≥0 of positive integers and a sequence of maps S n k : D 2 −→ X such that : 2 M. Amouch and H. Lakrimi (i) T n k x S n k y −→ 0 for every x ∈ D 1 and y ∈ D 2 ; (ii) T n k S n k y −→ y for every y ∈ D 2 .
For a more general overview of hypercyclicity, supercyclicity and related properties in linear dynamics, we refer to [1,2,7,12,18,19]. For T ∈ B(X), denote by L T and R T the left multiplication operator defined by L T (S) = T S, for S ∈ B(X) and the right multiplication defined by R T (S) = ST , for S ∈ B(X), respectively. Recall [11] that (J, . J ) is said to be a Banach ideal of B(X) if : (ii) The norm . J is complete in J and S ≤ S J for all S ∈ J; (iii) ∀S ∈ J, ∀A, B ∈ B(X), ASB ∈ J and ASB J ≤ A S J B ; (iv) The one-rank operators x ⊗ x * ∈ J and x ⊗ x * J = x x * for all x ∈ X and x * ∈ X * .
A one-rank operator defined on X as (x ⊗ x * )(z) = z, x * x = x * (z)x for all x ∈ X, x * ∈ X * and any z ∈ X. The space of finite rank operators F(X) is defined as the linear span of the one-rank is an admissible Banach ideal of B(X), we denote by L J,T and R J,T the left multiplication operator defined by L J,T (S) = T S, for S ∈ J and the right multiplication defined by R J,T (S) = ST , for S ∈ J, respectively. Similarly, we denote by L K,T and R K,T the left and the right multiplication operators induced by T ∈ B(X) on K(X) endowed with the norm operator topology, respectively. Let Y and Z be two normed linear spaces. The projective tensor norm on Y Z is the function x j y j : z = n j=1 x j ⊗ y j }.
For z = x ⊗ y, we have Π(z) = x y , with this topology the space is denoted by Y π Z and its completion by Y π Z. For more general information about the projective tensor norm and its related properties we refer to [6,14,20]. In the setting of Banach ideals, J. Bonet et al in [4], use tensor product techniques to characterize the hypercyclicity of the left and the right multiplication operators on (J, . J ). Subsequently, Yousefi et al in [22], characterized the supercyclicity of the left multiplication using Hilbert Schmidt operators. In [13], Gupta et al gave a sufficient criterion for the map C A,B (S) = ASB to be supercyclic on certain algebras of operators on Banach space. Recently, Gilmore et al in [9,10], investigate the study of the hypercyclicity properties of the commutator maps L T − R T and the generalized derivations L A − R B on (J, . J ). In the present work, we will characterize the supercyclicity of the left and the right multiplication on (J, . J ), also we give a sufficient conditions for the tensor product T ⊗R of two operators to be supercyclic. As a consequence, we give some equivalent conditions for the supercyclicity criterion. In section 2, we study the passage of the property of being supercyclic from an operator T ∈ B(X) to L J,T and R J,T . So, we prove that : (i) T satisfies the supercyclicity criterion on X if and only if L J,T is supercyclic.
(ii) T * satisfies the supercyclicity criterion on X * if and only if R J,T is supercyclic.
In section 3, we give a sufficient conditions for the tensor product T ⊗R of two operators to be supercyclic. As a consequence, we prove that if Y and Z are two separable Banach spaces such that dim Z > 1 then T ∈ B(Y ) satisfies the supercyclicity criterion if and only if T ⊗I : Y ⊗ π Z −→ Y ⊗ π Z is supercyclic for the projective tensor norm π.

Supercyclicity of the left and the right multiplication on Banach ideal of operators
We begin this section with the following lemma which will be used in the sequel.
Lemma 2.1. Let (J, . J ) be an admissible Banach ideal of B(X). If D and Φ are a countable dense subsets of X and X * , respectively. Then the set Thus X is a dense subset of J with respect to . J -topology. ✷ In the setting of Banach ideals, J. Bonet et al [4] prove that : (i) T satisfies the hypercyclicity criterion on X if and only if L J,T is hypercyclic.
(ii) T * satisfies the hypercyclicity criterion on X * if and only if R J,T is hypercyclic.
In the following, we prove that this result holds for supercyclicity. Proof. ⇒) Assume that T satisfies the supercyclicity criterion on X, then there exist a strictly increasing sequence (n k ) k of positive integers, two dense subsets D 1 , D 2 of X and a sequence of maps S n k : Let Φ be a dense subset of X * and consider the sets Using the assumption a), we show that ( In the other hand Supercyclicity of Multiplication on Banach Ideal of Operators 5 we have : Using the assumption b), we prove that L n k J,T Q n k (B) − B J −→ 0, as k −→ +∞. Hence L J,T satisfies the supercyclicity criterion. Thus L J,T is supercyclic. ⇐) Suppose that L J,T is supercyclic. Assume that x 1 , x 2 ∈ X are linearly independent and define then ϕ is surjective. Indeed, let y 1 , y 2 ∈ X, by using the Hahn-Banach theorem, there exist Thus T ⊕ T is supercyclic on X X by quasi-similarity. Hence, [2, Lemma 3.1], implies that T satisfies the supercyclicity criterion. ✷ It was shown in [17,21] that the supercyclicity of L T on B(X) with the strong operator topology provided that T satisfies the supercyclicity criterion. In the following corollary, we give a simple and different proof for supercyclicity of L T from what has been proven in [17,21]. (ii) L K,T is supercyclic.
(iii) L T is supercyclic on B(X) in the strong operator topology.
Proof. (i) ⇔ (ii) Consequence of Theorem 2.2, since K(X) is an admissible Banach ideal of B(X). (i) ⇒ (iii) Suppose that T satisfies the supercyclicity criterion on X. Let U and V be two non-empty open subsets of B(X) in the strong operator topology. Since K(X) is dense in B(X) with the strong operator topology [5, Corollary 3], there exist A 1 , A 2 ∈ K(X) such that A 1 ∈ U and A 2 ∈ V . Thus we can find x 1 , x 2 ∈ X\{0} and ǫ 1 , ǫ 2 > 0 such that U i is a non-empty open subset of K(X) with the norm operator topology. By Theorem 2.2 with J = K(X), L K,T is supercyclic, so there is some α ∈ C and n ∈ N such that Hence, it follows that α(L T ) n U ∩V = ∅. Thus, L T is supercyclic on B(X) in the strong operator topology. Proof. ⇒) Assume that T * satisfies the supercyclicity criterion on X * , then there exist a strictly increasing sequence (n k ) k of positive integers, two dense subsets Φ 1 , Φ 2 of X * and a sequence of maps M n k : Let D be a dense subset of X and consider the sets Φ 0 := span{x ⊗ ϕ/x ∈ D, ϕ ∈ Φ 1 } and Ψ 0 := span{y ⊗ φ/y ∈ D, φ ∈ Φ 2 } and the maps N n k : Ψ 0 −→ J define by By Lemma 2.1, Φ 0 and Ψ 0 are subsets of J which are . J -dense in J.
Using the assumption a), we show that (R J,T ) n k A J N n k B J −→ 0, as k −→ +∞. In the other hand we have Using the assumption b). we prove that (R J,T ) n k N n k (B) − B J −→ 0, as k −→ +∞. Hence R J,T satisfies the supercyclicity criterion. Thus R J,T is supercyclic. ⇐) Suppose that R J,T is supercyclic. Let x * 1 , x * 2 ∈ X * are linearly independent and define Then φ is surjective, indeed, let y * 1 , y * 2 ∈ X * , we take x * 1 , x * 2 ∈ X * such that x * i (x j ) = δ i,j and set

M. Amouch and H. Lakrimi
For A ∈ J, we have Thus T * ⊕ T * is supercyclic on X * X * . Hence, by [2, Lemma 3.1], T * satisfies the supercyclicity criterion on X * . ✷ Corollary 2.5. If K(X) is an admissible Banach ideal of B(X), then for all T ∈ B(X), the following statements are equivalent : (i) T * satisfies the supercyclicity criterion on X * .
(iii) R J,T is supercyclic on B(X) in the strong operator topology.
Proof. (i) ⇔ (ii) Consequence of Theorem 2.4, since K(X) is an admissible Banach ideal of B(X).
(i) ⇒ (iii) Suppose that T * satisfies the supercyclicity criterion on X * . Let U and V be two non-empty open subsets of B(X) in the strong operator topology. Since K(X) is dense in B(X) with the strong operator topology [5, Corollary 3], there exist A 1 , A 2 ∈ K(X) such that A 1 ∈ U and A 2 ∈ V . Thus we can find x 1 , x 2 ∈ X\{0} and ǫ 1 , ǫ 2 > 0 such that U i is a non-empty open subset of K(X) with the norm operator topology. By Theorem 2.4 with J = K(X), R J,T is supercyclic on (K(X), ||.||), so there is some α ∈ C and n ∈ N such that Hence, it follows that α(R J,T ) n U ∩ V = ∅. Thus R J,T is supercyclic on B(X) in the strong operator topology.
(iii) ⇒ (i) By the same technique as in the proof of Theorem 2.4. ✷

Tensor stability of supercyclicity
In [16] the authors gave a sufficient conditions for the tensor product T ⊗R of two operators to be hypercyclic. We extend this results to the supercyclic case, we give a sufficient conditions for the tensor product T ⊗R of two operators to be supercyclic.
Definition 3.1. Let T ∈ B(X). We say that T satisfies the tensor supercyclicity criterion if there exists two dense subsets D 1 , D 2 ⊂ X, an increasing sequence (n k ) k∈N of positive integers, (λ n k ) k∈N ⊂ C\{0} and a sequence of maps S n k : D 2 −→ X such that : Supercyclicity of Multiplication on Banach Ideal of Operators 9 (iii) (T n k S n k y) k∈N −→ y for all y ∈ D 2 . Example 3.2.
1. A sequences of operators satisfying the supercyclicity criterion satisfy the tensor supercyclicity criterion .
2. The identity map on X satisfies the tensor supercyclicity criterion . 3. Any isometry on a Banach space satisfies the tensor supercyclicity criterion with respect to the sequence of all positive integers.
Theorem 3.3. Let Y and Z be two separable Banach spaces. If T 1 ∈ B(Y ) satisfies the supercyclicity criterion and T 2 ∈ B(Z) satisfies the tensor supercyclicity criterion with respect to the same sequence (n k ) k of positive integers, then satisfies the supercyclicity criterion.
, for all k ∈ N, linear maps satisfying the conditions of supercyclicity criterion and tensor supercyclicity criterion for T 1 ∈ B(Y ) and T 2 ∈ B(Z), respectively. We will see that X 1 := D 1 ⊗ D 3 , X 2 := D 2 ⊗ D 4 , (λ n k := λ 1 n k .λ 2 n k ) k∈N and the maps S n k := S 1 n k ⊗ S 2 n k : X 2 −→ Y ⊗ Z are such that the conditions of the supercyclicity criterion are satisfied for the operator Indeed, for every x 1 ∈ D 1 , x 2 ∈ D 3 , y 1 ∈ D 2 and y 2 ∈ D 4 : Finally, which completes the proof by taking a linear combinations of elementary tensors. ✷ The following Corollary is an immediate consequence of the above Theorem.
In the following Corollary, we show the connection between supercyclicity of tensor products and supercyclicity of direct sum, and yields another equivalent formulation in the context of tensor products of the supercyclicity criterion.
Corollary 3.5. Let Y and Z be two separable Banach spaces with dim Z ≥ 2 and T ∈ B(Y ). The following are equivalent : (i) T satisfies the supercyclicity criterion.
Thus T ⊗I is quasi-similar to T ⊕ T , hence T ⊕ T is supercyclic on Y Y . ✷ Remark 3.6. All the results in this section are valid for any tensor norm a see [6], since the π-topology is the finest one.