Paris proved in 2001 Herrero's conjectured "that every multi-hypercyclic (multi- supercyclic) operator on a Hilbert space is in fact hypercyclic (supercyclic)&nbsp; In this paper we study this conjectured in many cases of operators between hypercyclic operators and supercyclic operators. And we proved that multi G-cyclic operator is G-cyclic operator

Jamil, Zeana Z.

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ISSN 2347-1921                                                           1632 | P a g e                                                             M a y  1 9 , 2 0 1 4  Multi G –cyclic Operators are G-cyclic Zeana Z.  Jamil University of Baghdad – College of Science – Dept. of Mathematics Zeana.zaki@scbaghdad.edu.iq  ABSTRACT Paris proved in 2001 Herrero’s conjectured “that every multi-hypercyclic (multi- supercyclic) operator on a Hilbert space is in fact hypercyclic (supercyclic)”.  In this paper we study this conjectured in many cases of operators between hypercyclic operators and supercyclic operators. And we proved that multi -cyclic operator is -cyclic operator. Keywords Hypercyclic operators; supercyclic operators. SUBJECT  CLASSIFICATION  Mathematics Subject Classification: 47A16, 47B37.                          Council for Innovative Research Peer Review Research Publishing System Journal: Journal of Advances in Mathematics Vol 8, No 3 editor@cirjam.org www.cirjam.com, www.cirworld.com  ISSN 2347-1921                                                           1633 | P a g e                                                             M a y  1 9 , 2 0 1 4  INTRODUCTION: Let  be an infinite dimensional separable complex Hilbert space, and let  be the Banach algebra of all bounded linear operators on . Let , it is called hypercyclic if there is a vector  in  such that the set  is norm–dense in  [1]. An operator  is supercyclic if there exists an   such that is norm–dense in  [1].  is called multi-hypercyclic if there is a set of vectors  in  such that the set  is norm–dense in  [2]. An operator  is multi-supercyclic if there exists a set of vectors  in  such that is norm–dense in  [2]. In 1999, Herrero conjectured that every multi-hypercyclic (multi-supercyclic) operator is hypercyclic (supercyclic) [1]. Peris in 2001 settle this conjecture in the affirmative even for continuous linear operator [2]. In 2005, Naoum and Jamil introduced -cyclic operator, an operator  is called -cyclic over a multiplication semigroup of  with , if there exists a vector  such that  is norm dense in , such a vector is called -cyclic vector for  over  [3]. It is easy to see that every hypercyclic operator is - cyclic operator, and every - cyclic operator is supercyclic In this paper we introduce the concept multi -cyclic operator,  is called multi -cyclic operator if there exists a finite set of vectors  in  such that is norm–dense in . We prove that every multi -cyclic operator is -cyclic oprerator. Main result: First we need the following lemmas. Since every -cyclic operator is supercyclic then we get: Lemma 1: If  is multi -cyclic, then  has at most one eigenvalue  Lemma 2: Let , then  if and only if . Proof: ) Clearly ) Since , we find ,  such that . Since  is –invariant, then , for all . Thus  By the same argument we get the other inclusion. Before we prove our main result, we list without proof some facts which will be frequently used in the proof. Observations 3: 1. Let  be a topological space and  a finite family of closed subsets of  such that . If  then . 2. Let be a topological space, , and  a closet subset of  with . Then . 3. If  is an complex Housdorf local convex space  and us a proper subspace, then  is connected. 4. Any subspace of containing a non-empty open set must be all of . 5. Let  be a complex polynomial,  has dense range if and only if  for every eigenvalue  of . ISSN 2347-1921                                                           1634 | P a g e                                                             M a y  1 9 , 2 0 1 4  6.  Now we are ready to state and prove the main theorem. Theorem 4: If  is multi -cyclic operator, then  is -cyclic operator. Proof Since  is multi -cyclic, then there exists a finite set  such that  and  is minimal. We define . If ,  is -cyclic. Now, if , then by ((3), part (1)) and the minimality of ,  for all ; . By (2) given any  with  is not empty.  for some ; .                                    (1) Moreover,                                                        (2) Otherwise, there are ,  such that  for some . And by ((3), part (2)), , thus , a contradiction. Since  has one possible eigenvalue, say , then let  be a polynomial in  such that . By ((3), part (5))  has dense range, thus . Since  and the minimality of , then  for all ; . So by , with  and . Then . Now observe that . Thus by ((3), part (3))  is connected, hence . By ((3), part (4))  is dense, then  Therefore  is -cyclic operator. References: [1] D. A. Herrero,Hyperc yclic operators and chaos,J. Operator Theory 28 (1992),pp. 93–103 [2] Peris, Multi-hypercyclic operators are hypercyclic, Math. Z. 236, (2001), pp. 779–786  [3] G. Naoum, Z. Z. Jamil, G-cyclicity, Journal of Al-Nahrain University, 8(2), (2005), pp. 103-108.  

Multi G -cyclic Operators are G-cyclic

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Multi G -cyclic Operators are G-cyclic

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