Cyclic Direct Sum of Tuples

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 8, Cyclic Direct Sum of Tuples Bahmann Yousefi Fariba Ershad Department of Mathematics, Payame Noor University P.O. Box: , Shiraz, Iran b yousefi@pnu.ac.ir, fershad@pnu.ac.ir Abstract In this paper we characterize conditions for the direct sum of a tuple of operators to be cyclic. Mathematics Subject Classification: 47B37; 47B33 Keywords: tuple of operators, hypercyclic vector, cyclic vector 1 Introduction Let T =(T 1,T 2 ) be a pair of commuting continuous linear operators acting on an infinite dimensional Banach space X. We will let F = {T 1 k 1 T 2 k 2 : k i 0,i=1, 2}. For x X, the orbit of x under the tuple T is the set Orb(T,x)={Sx : S F}. A vector x is called a hypercyclic vector for T if Orb(T,x) is dense in X in this case the tuple T is called hypercyclic. Also, by T d we will refer to the set T d = {S S : S F}. We say that T d is hypercyclic provided there exist x 1,x 2 X such that {W (x 1 x 2 ):W T d } is dense in X X. Similarly, A vector x is called a cyclic vector for T if the linear span of the orbit Orb(T,x) is dense in X in this case the tuple T is called cyclic. By a polynomial p(.,.) we will mean m n p(z, w) = c ij z i w j. i=1 j=1

2 378 B. Yousefi F. Ershad Let P = {p(t 1,T 2 ):p(.,.) is a polynomial}, note that the linear span of the orbit Orb(T,x) is equal to {Sx : S P}. Thus the tuple T is cyclic if {Sx : S P}is dense in X. Also, let P(T ) d = {S S : S P}. We say that P(T ) d is cyclic if there exist x 1,x 2 X such that is dense in X X. {W (x 1 x 2 ):W P(T ) d } Hypercyclic operators arise within the class of composition operators ([4]), weighted shifts ([12]), adjoints of multiplication operators ([5]), adjoints of subnormal hyponormal operators ([2]), hereditarily operators ([1]), topologically mixing operators ([7, 11]). Here, we want to extend some properties of hypercyclic cyclic operators to a pair of commuting operators, although the techniques work for any n-tuple of operators but for simplicity we prove our results only for the case n = 2. For some other topics we refer to [1 16]. 2 Main Results In this section we characterize some conditions for the direct sum of a pair of operators to be cyclic. We will denote the collection of hypercyclic cyclic vectors of a pair T by HC(T ) C(T ), respectively. Lemma 2.1 Let X be a separable infinite dimensional Banach space T =(T 1,T 2 ) be a cyclic pair of operators T 1 T 2.IfTi for i =1, 2, then C(T ) is dense. Proof. Note that if x is a cyclic vector for T, then for all polynomials p(.,.) we have: span(orb(t,p(t 1,T 2 )x)) = {q(t 1,T 2 )p(t 1,T 2 )x : q(.,.) is a polynomial} = p(t 1,T 2 ){q(t 1,T 2 )x : q(.,.) is a polynomial}. Since Ti for i =1, 2, thus p(t 1,T 2 ) has dense range so the linear span of Orb(T,p(T 1,T 2 )x) is dense in X. Hence, for all polynomials p(.,.) so This implies that C(T ) is dense in X. p(t 1,T 2 )x C(T ) span(orb(t,x)) C(T ).

3 Cyclic direct sum of tuples 379 Theorem 2.2 Let X be a separable infinite dimensional Banach space T =(T 1,T 2 ) be the pair of operators T 1 T 2 Ti for i =1, 2. IfT is cyclic, then for any nonempty open subsets U, V of X, there exists a polynomial q(.,.) such that the set q(t 1,T 2 )(U) V is nonempty. Proof. Since C(T ) is dense, thus the set U C(T ) is nonempty. Choose x U C(T ). Since x is a cyclic vector, is dense in X so {p(t 1,T 2 )x : p(.,.) is a polynomial} V {p(t 1,T 2 )x : p(.,.) is a polynomial} is nonempty. Thus there exists a polynomial q(.,.) such that q(t 1,T 2 )x V C(T ), from which we can conclude that q(t 1,T 2 )(U) V is nonempty. Theorem 2.3 Let X be a separable infinite dimensional Banach space T =(T 1,T 2 ) be the pair of operators T 1 T 2 Ti for i =1, 2. If P(T ) d is cyclic, then for any nonempty open subsets U 1, U 2, V 1, V 2 of X, there exists a polynomial q(.,.) such that the sets q(t 1,T 2 )(U 1 ) V 1 q(t 1,T 2 )(U 2 ) V 2 are nonempty. Proof. By a similar method used in Lemma 2.1, one can see that P(T ) d is dense in X X. Now, by the technique used in the proof of Theorem 2.2, the proof is complete. Theorem 2.4 Let X be a separable infinite dimensional Banach space T =(T 1,T 2 ) be a hypercyclic pair of operators T 1, T 2, Ti for i =1, 2. If for every nonempty open subsets U, V of X every neighborhood W of 0, there exists a polynomial q(.,.) such that q(t 1,T 2 )(U) W Ø then T d is hypercyclic. q(t 1,T 2 )(W ) V Ø, Proof. It is sufficient to show that for every nonempty open subsets U, V of X every neighborhood W of 0, there exist integers m n such that T m 1 T n 2 (U) W Ø

4 380 B. Yousefi F. Ershad T1 m T 2 n (W ) V Ø. So let U, V be any nonempty open subsets of X W be a neighborhood of 0. Let q(.,.) be the polynomial such that q(t 1,T 2 )(U) W Ø q(t 1,T 2 )(W ) V Ø. Since T is hypercyclic, thus HC(T is dense in X so HC(T ) (q(t 1,T 2 ) 1 (W ) U) Ø. Choose x HC(T ) (q(t 1,T 2 ) 1 (W ) U), thus x U q(t 1,T 2 )x W. Since q(t 1,T 2 ) 1 V W Ø it is also open, we get (q(t 1,T 2 ) 1 V W ) Orb(T,x) Ø. Hence there exist integers m n such that T1 m T 2 n x q(t 1,T 2 ) 1 V W. Since x U T1 mt 2 nx W,so T1 m T 2 n (U) W Ø. Also, thus q(t 1,T 2 )T1 m T2 n x V q(t 1,T 2 )x W, T1 m T 2 n (W ) V Ø. Now the proof is complete.

5 Cyclic direct sum of tuples 381 Corollary 2.5 Let X be a separable infinite dimensional Banach space T =(T 1,T 2 ) be a hypercyclic pair of operators T 1, T 2, Ti for i =1, 2. Then for every nonempty open subsets U, V of X every neighborhood W of 0, there exists a polynomial q(.,.) such that if only if P(T ) d is cyclic. q(t 1,T 2 )(U) W Ø q(t 1,T 2 )(W ) V Ø, Corollary 2.6 Let X be a separable infinite dimensional Banach space T =(T 1,T 2 ) be a hypercyclic pair of operators T 1, T 2, Ti for i =1, 2. Then T d is hypercyclic if only if P(T ) d is cyclic. References [1] J. Bes A. Peris, Hereditarily hypercyclic operators, J. Func. Anal., 167 (1) (1999), [2] P. S. Bourdon, orbits of hyponormal operators, Mich. Math. Journal, 44 (1997), [3] P. S. Bourdon N. Feldman, Somewhere dense orbits are everywhere dense, Indiana Univ. Math. Journal, 52 (2003), [4] P. S. Bourdon J. H. Shapiro, Cyclic phenomena for composition operators, Memoirs of the Amer. Math. Soc., 125, Amer. Math. Soc. Providence, RI, [5] P. S. Bourdon J. H. Shapiro, Hypercyclic operators that commute with the Bergman backward shift, Trans. Amer. Math. Soc., 352 (2000), [6] K. C. Chan, Hypercyclicity of the operator algebra for a separable Hilbert space, J. Operator Theory, 42 (1999), [7] G. Costakis M. Sambarino, Topologically mixing hypercyclic operators, Proc. Amer. Math. Soc., 132 (2003), [8] N. S. Feldman, Hypercyclic pairs of coanalytic Toeplitz operators, Integral Equations Operator Theory, 58 (20) (2007), [9] N. S. Feldman, Hypercyclic tuples of operators somewhere dense orbits, J. Math. Appl., 346 (2008),

6 382 B. Yousefi F. Ershad [10] G. Godefroy J. H. Shapiro, operators with dense invariant cyclic manifolds, J. Func. Anal., 98 (1991), [11] S. Grivaux, Hypercyclic operators, mixing operators the bounded steps problem, J. Operator Theory, 54 (2005), [12] H. N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc., 347 (1995), [13] B. Yousefi, H. Rezaei, Hypercyclicity on the algebra of Hilbert- Schmidt operators, Results in Mathematics, 46 (2004), [14] B. Yousefi A. Farrokhinia, On the hereditarily hypercyclic vectors, Journal of the Korean Mathematical Society, 43 (6) (2006), [15] B. Yousefi H. Rezaei, Hypercyclic property of weighted composition operators, Proc. Amer. Math. Soc., 135 (10) (2007), [16] B. Yousefi H. Rezaei, On the supercyclicity hypercyclicity of the operator algebra, Acta Mathematica Sinica, 24 (7) (2008), Received: November, 2010