Hypercyclic Functions for Backward and Bilateral Shift Operators

Problem statement: Giving conditions for bilateral forward and unilateral bac kward shift operators over the weighted space of p-summable for mal series to be hypercyclic. This provides a generalization to the case of Hilbert space. Approach: We used hypercyclicity criterion and some preliminary concepts for formal Laurent series and formal power series. Moreover we got benefits of some duality properties of above mentioned spaces. R ults: We obtained necessary and sufficient conditions for bilateral forward and unilateral bac kward shift operators to be hypercyclic. Conclusion: The bilateral forward shift operator was hypercycli c on the space of all formal Laurent series and the unilateral backward shift operator w as hypercyclic on the space of all formal power series under certain conditions.


INTRODUCTION
A vector x in a Banach space X is called hypercyclic for a bounded operator T if the orbit { } n T x : n 0 ≥ is dense in X. The first examples of hypercyclic operators appeared in the space of entire functions defined over the complex plane, endowed with the compact-open topology. In 1929 Birkhoff [1] essentially showed the hypercyclicity of the translation operators a T f (z) f (z a),a 0 = + ≠ , while MacLane [2] proved the hypercyclicity of the differentiation operator. The notion of hypercyclicity on Banach spaces started in 1969 with Rolewicz [3] , who showed that any scalar multiple λ B of the unilateral backward shift B is hypercyclic on p l (1 p< ) ≤ ∞ and c 0 , whenever 1 λ > . Kitai in his thesis with title invariant closed sets for linear operators, university of Toronto, determined conditions under which a linear operator is hypercyclic. This result, commonly referred to as the hypercyclicity criterion, was never published and a few years later it was rediscovered in a broader form by Gethner and Shapiro [5] . During the last year's hypercyclicity criterion on Banach or Frechet spaces has attracted many mathematicians working in linear functional analysis and very important contributions to the topic have been made [5][6][7][8][9][10][11][12][13] . We use the hypercyclicity criterion of [10] G. Godefroy and J. H. Shapiro (1991), to show that the bilateral forward shift operators on the space of all formal Laurent series and the unilateral backward shift operators on the space of all formal power series are hypercyclic. Our results in the case 2 p = are compatible to that given by H. Salas (1995) on the space l 2 (Z) and generalize those given by [12] . is called the bilateral forward shift; furthermore, the inverse of M z which is the bilateral backward shift is the operator B z defined on P

MATERIALS AND METHODS
Noting that the shift operator defined over the  Without loss of generality [4] , we may assume that the    (2) is sufficient for the convergence of the series kq q k z (k) β ∑ for every z, then by using Hölder inequality we get: This is the required result. One can easily see that the dual of p E β is q E γ , where

RESULTS
Now we give necessary and sufficient conditions for bilateral forward shift operator on the space p E β is hypercyclic.
Proof: Assume that z M is hypercyclic. Since the set of hypercyclic functions for M z is dense, then given ε > 0 and q∈N, let 0 1 < δ < so that 0 / (1 ) < δ − δ < ε and there is a hypercyclic function f(z) for M z such that: Thus the hypercyclicity criterion is satisfied and the proof is complete.

DISCUSSION
Results obtained from proposition 5 can be applied in the next proposition but for the unilateral backward shift operator on the space p H β .  (8) Proof: Assume that S z is hypercyclic. Since the set of hypercyclic functions for S z is dense, then given ε > 0 and q∈N, let 0 Thus the hypercyclicity criterion is satisfied and the proof is complete.

CONCLUSION
The bilateral forward shift operator is hypercyclic on the space of all formal Laurent series and the unilateral backward shift operator is hypercyclic on the space of all formal power series under certain conditions. Propositions 5 and 6 in the case of p 2 = are compatible with that given in [7] over the space l 2 (Z) and in [10] over the space l 2 (N). Noting that, some examples of hypercyclic bounded linear operators have applications in physics and quantum radiation field theory [10,12] .