Some Results on Subspace Cesaro-Hypercyclic Operators

. In this paper we characterize the notion of subspace Ces` a ro-hypercyclic. At the same time, we also provide a Subspace Ces` a ro-hypercyclic Criterion and o ﬀ er an equivalent conditions of this criterion.


Introduction
Let H be a separable infinite dimensional Hilbert space over the scalar field C. As usual, N is the set of all non-negative integers, Z is the set of all integers, and B(H ) is the space of all bounded linear operators on H.A bounded linear operator T : H → H is called hypercyclic if there is some vector x ∈ H such that Orb(T, x) = {T n x : n ∈ N} is dense in H, where such a vector x is said hypercyclic for T.
The first example of hypercyclic operator was given by Rolewicz in [16].He proved that if B is a backward shift on the Banach space l p , then λB is hypercyclic if and only if |λ| > 1.
Let {e n } n≥0 be the canonical basis of l 2 (N).If {w n } n∈≥1 is a bounded sequence in C\{0}, then the unilateral backward weighted shift T : l 2 (N) −→ l 2 (N) is defined by Te n = w n e n−1 , n ≥ 1, Te 0 = 0, and let {e n } n∈Z be the canonical basis of l 2 (Z).If {w n } n∈Z is a bounded sequence in C\{0}, then the bilateral weighted shift T : l 2 (Z) −→ l 2 (Z) is defined by Te n = w n e n−1 .The definition and the properties of supercyclicity operators were introduced by Hilden and Wallen [9].They proved that all unilateral backward weighted shifts on a Hilbert space are supercyclic.
A bounded linear operator T ∈ B(H ) is called supercyclic if there is some vector x ∈ H such that the projective orbit C.Orb(T, x) = {λT n x : λ ∈ C, n ∈ N} is dense in X.Such a vector x is said supercyclic for T. Refer to ( [1], [8], [4], [19]) for more informations about hypercyclicity and supercyclicity.
A nice criterion namely Hypercyclicity Criterion, was developed independently by Kitai [11] and, Gethner and Shapiro [7].The Hypercyclicity Criterion has been widely used to show that many different types of operators are hypercyclic.For instance hypercyclic operators arise in the classes of composition operators [3], adjoints of multiplication operators [7], cohyponormal operators [6], and weighted shifts [17].
Theorem 1.1.(Hypercyclicity Criterion).Suppose that T ∈ B(H ).If there exist two dense subsets X 0 and Y 0 in H and an increasing sequence n j of positive integer such that: (1) T n j x → 0 for each x ∈ X 0 , and (2) there exist mappings S n j : Y 0 −→ H such that S n j y → 0, and T n j S n j y → y for each y ∈ Y 0 , then T is hypercyclic.
In [17] and [18], Salas characterized the bilateral weighted shifts that are hypercyclic and those that are supercyclic in terms of their weight sequence.In [5], N. Feldman gave a characterization of the invertible bilateral weighted shifts that are hypercyclic or supercyclic.
Let M n (T) denote the arithmetic mean of the powers of T ∈ B(H ), that is If the arithmetic means of the orbit of x are dense in H then the operator T is said to be Ces àrohypercyclic.In [13], Fernando Le ón-Saavedra proved that an operator is Ces àro-hypercyclic if and only if there exists a vector x ∈ H such that the orbit {n −1 T n x} n≥1 is dense in H and characterized the bilateral weighted.The following examples give an operator which is Ces àro-hypercyclic but not hypercyclic and vice versa.
Example 1.1.[13] Let T the bilateral backward shift with the weight sequence Then T is not hypercyclic, but it is Ces àro-hypercyclic.
Example 1.2.[20] Let T the bilateral backward shift with the weight sequence Then T is not Ces àro-hypercyclic, but it is hypercyclic and supercyclic.
In 2011, B. F. Madore and R. A. Martnez-Avendano in [14] introduced and studied the concept of subspace-hypercyclicity for an operator.An operator T is subspace-hypercyclic or M-hypercyclic for a subspace M of X, if there exists x ∈ X such that Orb(T, x) M is dense in M. Such a vector x is called a M-hypercyclic vector for T, they showed that there are operators which are Mhypercyclic but not hypercyclic.They introduced analogously the concept of subspace-transitivity.
Let T ∈ B(X) and M be a closed subspace of X, we say that T is M-transitive, if for any non-empty open sets U, V in M, there exists n ≥ 0 such that T −n (U) V contain a non-empty open subset of M.The authors showed that M-transitivity implies M-hypercyclicity.Note that the converse is not true, this is proven recently by C. M. Le in [12]; for more informations see ( [10], [15]).
Similarly, for subspace-supercyclicity, Zhao, Y.L. Sun and Y.H. Zhou in [21] provided a Subspace-Supercyclicity Criterion and offered two necessary and sufficient conditions for a path of bounded linear operators to have a dense G δ set of common subspace-hypercyclic vectors and common subspace-supercyclic vectors and they also constructed examples to show that subspacesupercyclic is not a strictly infinite dimensional phenomenon and that some subspace-supercyclic operators are not supercyclic.
In this present paper, we will partially characterize the notion of subspace Ces àro-hypercyclic.At the same time, we also provide a Subspace Ces àro-hypercyclic Criterion and offer an equivalent conditions of this criterion.

Main results
We will assume that the subspace M ⊂ H is topologically closed.We start with our main definitions.Definition 2.1.Let T ∈ B(H ) and M be a closed subspace of H.We say that T is M-ces àro-hypercyclic if there exists a vector x ∈ H such that Orb(T, x) M = {n −1 T n x : n ≥ 1} M is dense in M. We call x a M-ces àro-hypercyclic vector.
Theorem 2.2.Let T be a subspace ces àro-hypercyclic and M be a nonzero subspace of H. Then the following conditions are equivalent: (1) For every non-empty open U and V of M, there exist n ≥ 1 such that n −1 T n −1 (U) V contains a non-empty open subset of M.
(2) For every non-empty open U and V of M, there exist n ≥ 1 such that n −1 T n −1 (U) V is non-empty and n −1 T n (M) ⊂ M.
(3) For every non-empty open U and V of M, there exist n ≥ 1 such that n Proof.(1) ⇒ (2).Let U and V be tow nonempty open subsets of M. By (1) there exist n ≥ 1 such Next, We prove that n −1 T n (M) ⊂ M.
W is open of M then for all r enough small we have (2) ⇒ (3).Let U and V be nonempty open subsets of M, by (2) there exist n ≥ 1 such that At last, we see that the implication (3) ⇒ (1) is obvious and this completes the whole proof of the theorem.
Corollary 2.1.Let T be a subspace ces àro-hypercyclic and M be a nonzero subspace of H.If any of the conditions in theorem 2.2 is satisfied, then CH(T, M) is a dense subset of M.
Proof.We may assume the condition (3) in theorem 2.2 is satisfied, then for every non-empty open U of M and for all k ≥ 1, there exist n ≥ 1 such that the set n −1 T n −1 (U) B k is nonempty and open.Hence the set is nonempty and open.Furthermore, U ∩ A k ∅ for all k ≥ 1.Thus each A k is dense in M and so by the Baire category theorem and theorem 2.1 CH(T, M) is also dense in M.
Next we get the following theorem, which is the Subspace Ces àro-Hypercyclic Criterion, which is similar to the Supercyclictiy Criterion that was stated in [2]; see also [8].
Theorem 2.3.(Subspace Ces àro-Hypercyclic Criterion ) Let T be a subspace ces àro-hypercyclic and M be a nonzero subspace of H. Assume that there exist M 0 and M 1 , dense subsets of M, an increasing sequence (n k ) of positive integer and a sequence of mappings S n k : M is an invariant subspace for n −1 k T n k for all n ≥ 1.Then T is subspace ces àro-hypercyclic for M.
Proof.Let U and V be non-empty open subsets of M. By Theorem 2.2, it is enough to prove that there exist n ≥ 1 such that Since M 0 and M 1 are dense in M, there exist x ∈ M 0 ∩ V, y ∈ M 1 ∩ U.And since U and V are nonempty open subsets, there exists ε > 0 such that B M (x, ε) ⊆ V and B M (y, ε) ⊆ U.By assumption, there exist (n k ) such that we have that n −1 k T n k u ∈ U. Then (n −1 k T n k ) −1 (U) V ∅ and T is subspace ces àro-hypercyclic for M.
(2) (Outer Subspace Ces àro-Hypercyclic Criterion ) There exist an increasing sequence (n k ) of positive integer , a dense linear subspace Y 0 ⊆ M and, for each y ∈ Y 0 , a dense linear subspace X 0 of M such that: (a) There exists a sequence of mappings S n k : Proof.It is obvious that any operator satisfying the Subspace Ces àro-Hypercyclic Criterion also satisfie the criteria of (2).It suffices to show that (2) implies (1).Let U i , V i ⊆ M non-empty open sets with i = 1, 2. The same argument as in the proof of Theorem 3.2 in [2] can be used to show that there exist (n k ) of positive integer such that Then we can know that (T ⊕ T) is subspace ces àro-hypercyclic for M ⊕ M and (x, y) is subspace ces àro-hypercyclic vector for (T ⊕ T).In particular, x be subspace ces àro-hypercyclic vector for T and CH(T, M) M is a dense G δ subset of M. Let (U k ) be a base of 0-neighborhoods in M. Then there exist (n k ) of positive integer such that Hence (1) holds.We complete the proof.Proposition 2.1.Let T ∈ B(H ) satisfy the subspace ces àro-hypercyclicity criterion with respect to a sequence (n k ) .Then T is subspace ces àro-mixing.
Proof.We show that T is subspace ces àro-mixing.Let M 0 and M 1 be dense sets in M, that are for all n ≥ n 0 .Then Therefore, T and S are M 1 -ces àro-mixing and M 2 -ces àro-mixing operators, respectively.

Remark 2 . 1 .Example 2 . 1 .Theorem 2 . 1 .
The definition above reduces to the classical definition of ces àro-hypercyclic if M = H and we may assume that the subspace ces àro-hypercyclic vector x ∈ M, if needed.Let T be a ces àro-hypercyclic operator on H with ces àro-hypercyclic vector x and let I be the identity operator on H. Then the operator T ⊕ I : H ⊕ H → H ⊕ H is subspace ces àro-hypercyclic for the subspace M := H ⊕ {0} with the subspace ces àro-hypercyclic vector x ⊕ {0}, but T ⊕ I is not ces àro-hypercyclic on the space H ⊕ H. Let T be a subspace ces àro-hypercyclic and M be a nonzero subspace of H. ThenCH(T, M) = k≥1 n≥1 n −1 T n −1 (B k ),where (B k ) k≥1 is a countable open basis for the relative topology of M as a subspace of H.Proof.Let (B k ) k≥1 is a countable open basis for the relative topology of M as a subspace of H.We have x ∈ CH(T, M) if and only if {n −1 T n x : n ≥ 1} M is dense in M if and only if for each k ≥ 1,

Theorem 2 . 5 .
given in the subspace ces àro-hypercyclicity criterion.Let U and V are two nonempty opensets in M, then choose x ∈ M 0 ∩ V and y ∈ U ∩ M 1 and ε > 0 such that B(x, ε) ⊂ V and B(y, ε) ⊂ U.By Theorem 2.3, there exist k 0 ∈ N * so that for all k ≥ k 0 , || T n k n k x|| ≤ ε, ||S n k (y)|| ≤ ε, and || T n k n k S n k (y) − y|| ≤ ε.Then for each k ≥ k 0 we have z k = x + S n k y ∈ B(x, ε) ⊂ V and T n k n k z k ∈ B(y, ε) ⊂ U.That is, T n k n k (V) ∩ U ∅, ∀k ≥ k 0 .Hence T is subspace ces àro-mixing.Let T and S in B(H ) and M 1 , M 2 be a nonzero closed subspaces of H and (T ⊕ S) is (M 1 ⊕ M 2 )-ces àro-mixing operator, then T and S are M 1 -ces àro-mixing and M 2 -ces àro-mixing operators, respectively.Proof.let U 1 and U 2 be open sets in M 1 , and V 1 andV 2 be open sets in M 2 , then U 1 ⊕ V 1 and U 2 ⊕ V 2are open in M 1 ⊕ M 2 .So there exists an n 0 ≥ 1 such thatn −1 (T ⊕ S) −1