On Syndetically Hypercyclic Tuples

Transcription

1 Iteratioal Mathematical Forum, Vol. 7, 2012, o. 52, O Sydetically Hypercyclic Tuples Mezba Habibi Departmet of Mathematics Dehdasht Brach, Islamic Azad Uiversity, Dehdasht, Ira P. O. Box , Lidigo, Stockholm, Swede habibi.m@iaudehdasht.ac.ir Abstract I this paper we will give some coditios for a tuple of operators or tuple of weighted shifts to be Sydetically Hypercyclic. Mathematics Subject Classificatio: 47A16, 47B37 Keywords: Sydetically tuple, Hypercyclic tuple, Hypercyclic vector, Hypercyclicity Criterio, Sydetically Hypercyclic 1 Itroductio Let F be a topological vector space(tvs) ad T 1,,...,T are cotiuous mappig o F, ad T =(T 1,,..., T ) be a tuple of operators T 1,,...,T.The tuple T is weakly mixig, if ad oly if, For ay pair of o-empty ope subsets U,V i X, ad for ay sydetic sequeces {m k,1 }, {m k,2 },..., {m k, } with Sup k ( k+1,j k,j ) < for j =1, 2,...,, the there exist m k,1,m k,2,..., m k, such that U V, also, if ad oly if, it suffices i previous coditio, to cosider oly those sequeces m k,1,m k,2,..., m k, for which there is some m 1 1, m 2 1,...,m 1 with m k,1 {m j, 2m j } for all k ad all j. Reader ca see [1 10] for some iformatio. 2 Mai Results Let X be a metrizable ad complete topological vector space(f space) ad T =(T 1,,..., T ) is a -tuple of operators, the we will let F = {T 1 k 1 k 2...T k : k i 0} be the semigroup geerated by T.Forx X we take Orb(T,x)={Sx : S F}= {T 1 k 1 k 2...T k (x) :k i 0,i=1, 2,..., }.

2 2598 M. Habibi The set Orb(T,x) is called, orbit of vector x uder T ad Tuple T =(T 1,,..., T ) is called hypercyclic pair if the set Orb(T,x) is dese i X, that is Orb(T,x)={T 1 k 1 k 2...T k (x) :k i 0,i=1, 2,..., } = X. The tuple T =(T 1,,..., T ) is called topologically mixig if for ay give ope subsets U ad V of X, there exist positive umbers K 1, K 2,..., K such that T 1 k 1,i k 2,i...T,i k,i (U) V φ, k j,i K i, j =1, 2,..., A sequece of operators {T } 0 is said to be a hypercyclic sequece o X if there exists some x X such that its orbit is dese i X, that is Orb({T } 0,x)=Orb({x, T 1 x, x,...} = X I this case the vector x is called hypercyclic vector for the sequece {T } 0. Note that, if {T } 0 is a hypercyclic sequece of operators o X, the X is ecessarily separable. Also ote that, the sequece {T } is a hypercyclic sequece o X, if ad oly if, the operator T is hypercyclic operator o X. T is said to satisfy the Hypercyclicity Criterio if it satisfies the hypothesis of below theorem. Theorem 2.1 (The Hypercyclicity Criterio) Let X be a separable Baach space ad T =(T 1,,..., T ) is a -tuple of cotiuous liear mappigs o X. If there exist two dese subsets Y ad Z i X, ad strictly icreasig sequeces {m j,1 } j=1, {m j,2 } j=1,...,{m j, } j=1 such that: 1. T m j,1 1 T m j,2 2...T m j, 0 o Y as j, 2. There exist fuctios {S j : Z X} such that for every z Z, S j z 0, ad T m j,1 1 T m j,2 2...T m j, S j z z, the T is a hypercyclic -tuple. A strictly icreasig sequece of positive itegers { k } k is said to be sydetic sequece, if Sup k ( k+1 k ) <. A tuple T =(T 1,,..., T ) o a space X is called sydetically hypercyclic if for ay sydetic sequeces of positive itegers {m k,1 } k, {m k,2 } k,..., {m k, } k the sequece { } k

3 O sydetically hypercyclic Ttuples 2599 is hypercyclic, i other had, there is x X such that {T k x : k 0} is des i X, that is, { (x)} = X. Give cotiuous liear operators T 1,,...,T that is T 1,,..., T L(X), defied o a separable F -space X, also suppose that T =(T 1,,..., T )be a tuple of operators T 1,,...,T, The T satisfies the Hypercyclicity Criterio if ad oly if for ay strictly icreasig sequeces of positive itegers {m k,1 } k, { k,2 } k,..., {m k, } k such that Sup k (m k+1,j m k,j ) < for all j, the the sequece { } k is hypercyclic. Also, for each hypercyclic vector x X of T, there exists two strictly icreasig sequece {m k } k,{ k } k such that Sup k (m k+1,j k,j ) < for all j, ad { } k is somewhere dese, but ot dese i X, That is, the tuple T =(T 1,,..., T ) ad the sequece { } k do ot share the same hypercyclic vectors. Let F be a Frechet space ad T 1,,..., T are bouded liear operators o F, ad T =(T 1,,..., T )bea tuple of operators T 1,,..., T. The space F is called topologically mixig if for ay give ope sets U ad V, there exist positive umbers M 1,M 2,...,M such that T m i,1 1 T m i,2 2...T m i, (U) V φ, m i,j M j, i =1, 2,..., Notice that, If the tuple T satisfies the hypercyclic criterio for sydetic sequeces, the T is topologically mixig tuple o space F. Let V be a topological vector space(tvs) ad T 1,,..., T are bouded liear operators o V, ad T =(T 1,,..., T ) be a tuple of operators T 1,,..., T. The tuple T is called weakly mixig if T T... T : X X... X X X... X is topologically trasitive. Theorem 2.2 Let X be a topological vector space(tvs) ad T 1,,..., T are cotiuous mappig o X, ad T = (T 1,,..., T ) be a tuple of operators T 1,,..., T. The the followig are equivalet: (i). T is weakly mixig. (ii). For ay pair of o-empty ope subsets U,V i X, ad for ay sydetic sequeces {m k,1 }, {m k,2 },..., {m k, }, there exist m k,1,m k,2,..., m k, such that T m k,1 1 T m k,2 2...T m k, (U) (V )

4 2600 M. Habibi (iii). It suffices i (ii) to cosider oly those sequeces {m k,1 }, {m k,2 },..., {m k, } for which there is some m 1 1, m 2 1,...,m 1 with for all k ad for all j. m k,j {m j, 2m j } proof (i) (ii). Give {m k,1 }, {m k,2 },..., {m k, } ad U,V satisfyig the hypothesis of coditio (ii), take for all j ad the product map tims {}}{ T T... T : m j = Sup k {m k+1,j m k,j } tims {}}{ X X... X tims {}}{ X X... X is trasitive, The there is m k,1,m k,2,..., m k, i N such that (T m k,1 1 T m k,2 2...T m k, (U)) ((T m k,1 1 ) 1 (T m k,2 2 ) 1...(T m k, ) 1 (V )) m k,1 =1, 2,..., m, m k,2 =1, 2,..., m,..., m k, =1, 2,..., m so (T m k,1 +m k,1 1 T m k,2 +m k,2 2...T m k, +m k, (U)) (V )) m k,1 =1, 2,..., m, m k,2 =1, 2,..., m,..., m k, =1, 2,..., m By the assumptio o {m k,1 }, {m k,2 },..., {m k, },for all j, we have {m k,j : k N} { +1,+2,..., + m j } If for all j we select m k,j {m k,j : k N} { +1,+2,..., + m j } the we have T m k,1 1 T m k,2 2...T m k, (U) (V ), by this the proof of (i) (ii) is completed. The case (ii) (iii) is trivial. Case (iii) (i). Suppose that U, V 1, V 2 are o-empty ope subsets of X, the there are {m k,1 }, {m k,2 },..., {m k, } i N such that U V 1 U V 2. This will imply that T is weakly mixig. Sice (iii) is satisfied, the we ca take {m k,1 }, {m k,2 },..., {m k, } i N such that V 1 V 2

5 O sydetically hypercyclic Ttuples 2601 By cotiuity, we ca fid Ṽ1 V 1 ope ad o-empty such that Ṽ 1 V 2. Also there exist some m k,1,m k,2,..., m k, i N such that T m k,1 +η 1 1 T m k,2 +η T m k, +η U Ṽ1 for η j =0,m j we take m k,j = m k,j + η j, for all j, ideed we fid strictly icreasig sequeces of positive itegers m k,1,m k,2,..., m k, such that m k,j {m j, 2m j } for all j,ad Now we have So the set is a subset of U Ṽ1 =, k N T m k,1 +η 1 1 T m k,2 +η T m k, +η U Ṽ1 (T m k,1 1 T m k,2 2...T m k, U Ṽ1) (T m k,1 +η 1 1 T m k,2 +η T m k, +η U) ( Ṽ 1 ) the we have (U) (V 1 ) φ ad similarly (U) (V 2 ) φ ow, this is the ed of proof. Refereces [1] J. Bes ad A. Peris, Hereditarily hypercylic operators, Jour. Fuc. Aal., 1 (167) (1999), [2] M. Habibi, -Tuples ad chaoticity, It. Joural of Math. Aalysis, 6 (14) (2012), [3] M. Habibi, -Tuples of Bouded Liear Operators o Baach Space, It. Math. Forum, 7 (18) (2012), [4] M. Habibi ad F. Safari, -Tuples ad Epsilo Hypercyclicity, Far East Jour. of Math. Sci., 47 (2) (2010),

6 2602 M. Habibi [5] M. Habibi ad B. Yousefi, Coditios for a tuple of operators to be topologically mixig, It. Jour. of App. Math., 23(6) (2010), [6] A. Peris ad L. Saldivia, Sydetically hypercyclic operators, Itegral Equatios Operator Theory, 51, No. 2 (2005) [7] H. N. Salas, Hypercyclic weighted shifts, Tras. Amer. Math. Soc., 347 (1995), [8] B. Yousefi ad M. Habibi, Sydetically Hypercyclic Pairs, It. Math. Forum, 5 (66) (2010), [9] B. Yousefi ad M. Habibi, Hereditarily Hypercyclic Pairs, It. Jour. of App. Math., 24(2) (2011)), [10] B. Yousefi ad M. Habibi, Hypercyclicity Criterio for a Pair of Weighted Compositio Operators, It. Jour. of App. Math., 24 (2) (2011), Received: May, 2012