ORBIT OF TUPLE OF OPERATORS TENDING TO INFINITY

By an n-tuple of operators we mean a finite sequence of length n of commuting bounded linear operators on a Banach space X. Throughout, X denotes a infinite dimensional Banach space and B(X) denotes the Banach algebra of all bounded linear operators on X. The orbit of a points x ∈ X under an operator T ∈ B(X) is the sequence {T x : n = 0, 1, ...} . For T ∈ B(X), with r(T), σ(T ), σp(T ) and σap(T ) will denote the spectral radius, the spectrum, the point spectrum and the approximate point spectrum of T,respectively. Recall that σp(T ) is the set of all eigenvalues of T while σap(T ) is the set of λ ∈ σ(T ) for there which there is a sequence of unit vectors


Introduction
By an n-tuple of operators we mean a finite sequence of length n of commuting bounded linear operators on a Banach space X.
Throughout, X denotes a infinite dimensional Banach space and B(X) denotes the Banach algebra of all bounded linear operators on X.The orbit of a points x ∈ X under an operator T ∈ B(X) is the sequence {T n x : n = 0, 1, ...} .
For T ∈ B(X), with r(T), σ(T ), σ p (T ) and σ ap (T ) will denote the spectral radius, the spectrum, the point spectrum and the approximate point spectrum of T,respectively.Recall that σ p (T ) is the set of all eigenvalues of T while σ ap (T ) is the set of λ ∈ σ(T ) for there which there is a sequence of unit vectors (x n ) n≥1 such that T x n − λx n → 0, as n → ∞ any such sequence is called a sequence of almost eigenvectors for λ.
We are interested in the operators T ∈ B(X) for which there is x ∈ X whose Orb(T, x) tends strongly to infinity, i.e.
Obviously, if σ p (T ) contains a point λ with |λ| > 1, then for every corresponding nonzero vector x in the eigenspace Ker(T − λ), the orbit will tends strongly to infinity: In general, Ker(T − λ) is not dense in X(relative to the norm topology).In order to produce a dense set of vectors in X whose orbits under T tend strongly to infinity we have to look at the points in the approximate point spectrum which are not eigenvalues.
In [5] S. Mancevska gaves a complete proof of the following theorem (originally stated by B. Beauzamy [9, Theorem 2.A.5]):If T ∈ B(X) and the circle {λ ∈ C/ |λ| = r(T )} contains a point in σ(T ) which is not an eigenvalue for T, then for every positive sequence (α n ) n≥1 with n≥1 α n < +∞, in every open ball in X with radius strictly larger then n≥1 α n , there is x ∈ X satisfying for all n ≥ 1.Note that, if r(T ) > 1, then the space will contain a dense set of vectors x ∈ X with orbits under T tending strongly to infinity.
If the sets σ ap (T )\σ p (T ) and σ ap (S)\σ p (S) both have a non-empty intersection with the domain {λ ∈ C/ |λ| > 1} then, there is a dense set of vectors x ∈ X such that both the orbits Orb(T ; x) and Orb(S;x) tend strongly to infinity.
In [2] S. Mancevska and M. Orovrance considered some conditions under which, given a sequence of bounded linear operators (T i ) i≥1 on an infinitedimensional complex reflexive Banach space X, and they show that there is a dense set of vectors in X whose orbits under each T i tend strongly to infinity.Theorem 1.3.(see [2], Corollary 10) Let X an infinite dimensional reflexive Banach space.If (T i ) i≥1 is a sequence in B(X) for wich there is β > 0 such that r(T i ) > 1 + β for all i ≥ 1 then, there is a dense set D in X such that Orb(T i ; x) tend strongly to infinity for every x ∈ D and i ≥ 1.
If (T i ) i≥1 is a sequence in B(X) satisfying the following, weaker condition then the one in Theorem 0.3.
( * ) the space may still contain a dense set of vectors with orbits under each T i , i ≥ 1 tending strongly to infinity.

Main Results
Definition 1.1.Let T = (T 1 , T 2 , ..., T n ) be an n-tuple of operators acting on an infinite dimensional Banach space X. Let be the semi-group generated by T .For x ∈ X, the orbit of x under the tuple T is the set Orb(T , x) = {Sx : S ∈ F} .
Definition 1.2.The orbit of x under the tuple T tending to infinity if: as k i → ∞ with k i ≥ 0, for all i = 1, ..., n.
In this paper are considered some conditions under which, The orbit of x under the tuple T tending to infinity.For simplicity we state and prove our results for a pair that is a tuple with n= 2, and the general case follows by a similar method.Definition 1.3.An operator T is bounded from below if and only there exists a constant C > 0 such that: Theorem 1.4.Let X an infinite dimensional reflexive Banach space and T = (T 1 , T 2 ) be the pair of operators T 1 and T 2 .
Suppose that the fllowing conditions hold true: 1) T 1 and T 2 are bounded from below.
Then there exists x ∈ X such that the orbit of x under the pair T tends strongly to infinity.
Proof.Using condition 2 we may apply Theorem 1.2.Then, there exists a dense set of vectors x ∈ X such that both the orbits Orb(T 1 ; x) and Orb(T 2 ; x) tend strongly to infinity.
Hence T 1 and T 2 are bounded from below or, i.e.Orb(T, x) tend strongly to infinity.
Corollary 1.1.Let X an infinite dimensional reflexive Banach space and T = (T 1 , T 2 , ..., T n ) be the n-tuple of operators in B(X)bounded below for all i ≥ 1.
If there is x ∈ X such that the orbit of x under T i for all i ≥ 1 tend strongly to infinity then the orbit of x under the tuple T tend strongly to infinity.
Remark 1.1.The converse is also true, ie If there is x ∈ X such that the orbit of x under the tuple T tend strongly to infinity then the orbit of x under T i for all i ≥ 1 tend strongly to infinity.
Proof.T i are the commuting bounded linear operators then, Therefore, Orb(T j , x) → ∞ for all j ≥ 1.
Example 1.1.Let S be the unilateral forward shift on ℓ 2 (N): where {e n : n ∈ N} is the standard orthonormal basis for ℓ 2 (N).Given a sequence of positive numbers (a i ) i≥1 so that a i ≻ 1 for all i ≥ 1 and a i → 1 as i → ∞ and let T i is unilateral injective forward weighted shift and hence (see [10], Theorem 6): σ p (T i ) = ∅ and σ ap (T i ) = {λ ∈ C/ |λ| = a i } .
Obviously, (T i ) i≥1 satisfies the weaker condition (*) and there exists a dense set of vectors in ℓ 2 (N) with orbits in each T i tending strongly to infinity.Actually T n i x = (a i S) n x = a n i x → ∞, n → ∞, for all x = 0 and i ≥ 1.
Moreover, T i is bounded from below.Indeed: Therefore, by the use of Corollary 1.1, the orbit of x in the tuple (T 1 , T 2 , ..., T 3 ) tends strongly to infinity.