International Journal of Applied Mathematics————————————————————–Volume 23 No. 6 2010, 973-976

CONDITIONS FOR A TUPLE OF OPERATORS

TO BE TOPOLOGICALLY MIXING

M. Habibi1 §, B. Yousefi2

1Islamic Azad UniversityBranch of Dehdasht

P.O. Box 7571763111, Dehdasht, IRANe-mail: habibi.m@iaudehdasht.ac.ir

2Department of MathematicsPayame-Noor University

P.O. Box 19395-4697, Tehran, IRANe-mail: b yousefi@pnu.ac.ir

Abstract: In this paper we characterize the conditions for a tuple of commu-tative bounded linear operators on a Frechet space to satisfy the topologicallymixing property.

AMS Subject Classification: 47B37, 47B33Key Words: tuple of operator, hypercyclic vector, hypercylicity criterion,topologically mixing property

1. Introduction

By an n-tuple of operators we mean a finite sequence of length n of commutingcontinuous linear operators on a locally convex space X. If T = (T1, T2, ..., Tn)is an n-tuple of operators, then we say that

F = {T1k1T2

k2, ..., Tnkn : ki ≥ 0}

is the semigroup generated by T . For x ∈ X, the orbit of x under the tuple T isthe set orb(T, x) = {Sx : S ∈ F}. A vector x is called a hypercyclic vector forT if orb(T, x) is dense in X and in this case the tuple T is called hypercyclic.

Received: June 12, 2010 c© 2010 Academic Publications§Correspondence author

974 M. Habibi, B. Yousefi

For simplicity we state and prove our results for n = 2, and the general casefollows by a similar method.

Definition 1.1. A pair (T1, T2) is called topologically mixing if for anygiven open sets U and V , there exist two positive integers M and N such that

Tm1 T n

2 (U) ∩ V �= φ; ∀m ≥ M, ∀n ≥ N.

A topologically mixing operator satisfies a much stronger condition thatholds for every large m and n. Roughly speaking, the iterates of any open setbecome well spread throughout the space. A nice criterion, namely the Hyper-cyclicity Criterion, is used in the proof of our main theorem. It was developedindependently by Kitai ([8]), Gethner and Shapiro ([7]). This criterion wasused to show that hypercyclic operators arise within the class of compositionoperators ([4]), weighted shifts ([9]), adjoints of multiplication operators ([5]),and adjoints of subnormal and hyponormal operators ([3]), and Hereditarilyoperators ([2]), and topologically mixing operators ([6]). The formulation ofthe Hypercyclicty Criterion in the following theorem was given by J. Bes ([1]).For some other basics on the topics we refer to [1]-[10].

2. Main Results

A nice criterion, namely the Hypercyclicity Criterion is used in the proof of ourmain theorem.

Theorem 2.1. (The Hypercyclicity Criterion) Suppose F is a separableBanach space and T = (T1, T2) is a 2-tuple of continuous linear mappings onF . If there exist two dense subsets Y and Z in F , and two strictly increasingsequences {mj} and {nj} such that:

1. Tmj

1 Tnj

2 → 0 on Y as j → ∞,2. There exists function {Sj : Z → F} such that for every z ∈ Z, Sjz → 0,

and Tmj

1 Tnj

2 Sjz → z.Then T is a hypercyclic tuple.

Remember that a strictly increasing sequence of positive integers {nk} issaid to be syndetic if sup {nk+1 − nk} < ∞. The following theorem is a gener-alization of Theorem 1.1 in [6] for tuples.

Theorem 2.2. Let T = (T1,T2) be a 2−tuple on a Frechet space (X, d).Assume that T satisfies the Hypercyclicity Criterion for syndetic sequences{mk}k and {nk}k. Then T is topologically mixing.

CONDITIONS FOR A TUPLE OF OPERATORS... 975

Proof. Let U and V be any open sets in X. We will show that there aretwo positive integers M and N such that

Tm1 T n

2 (U) ∩ V �= φ , ∀m ≥ M , ∀n ≥ N.

Since the sequences {mk}k and {nk}k in the Hypercyclicity Criterion are syn-detic, there are some positive integers m and n such that for all j ≥ 0 wehave

mk+1 − mk ≤ m , nk+1 − nk ≤ n.

For i = 0, 1, 2, ...,m and j = 0, 1, 2, ..., n consider open sets Vi,j such that

T i1T

j2 (Vi,j) = V.

Let Y and Z be the dense sets that are given in the Hypercyclicity Criterion.Choose x ∈ U ∩ Y , and take ε > 0 such that ballB(x, ε) ⊂ U . Also, for eachi = 0, 1, ...,m and j = 0, 1, ..., n take zi,j ∈ Vi,j ∩ Z. Assume that ε is smallenough such that the ballB(zi,j, 2ε) ⊂ Vi,j. For simplicity we denote d(z, 0) by‖ z ‖. Let k0 be large enough such that for all k ≥ k0 and i = 0, 1, 2, ...,m andj = 0, 1, 2, ..., n:

‖Tmk1 T nk

2 (x)‖ ≤ ε,

‖Sk(zi,j)‖ < ε,

‖Tmk1 T nk

2 Sk(zi,j) − zi,j‖ < ε.

Set M = mk0 and N = nk0. Let p ≥ M and q ≥ N . Then there are some mk

and nk with k ≥ k0 and 0 ≤ r ≤ m and 0 ≤ s ≤ n such that

p = mk + r and q = nk + s.

Define xp,q = x + Sk(zr,s). Then

‖ xp,q − x ‖=‖ Sk(zr,s) ‖< ε.

Hence xp,q ∈ B(x, ε) ⊂ U . Also, we have

T p1 T q

2 (xp,q) = T r1 T s

2 (Tmk1 T nk

2 (x + Sk(zr,s)).

Note thatTmk

1 T nk2 (x + Sk(zr,s)) ∈ V,

because

‖ Tmk1 T nk

2 (x + Sk(zr,s)) − zr,s ‖≤‖ Tmk

1 T nk2 (x) ‖ + ‖ Tmk

1 T nk2 Sk(zr,s) − zr,s ‖< 2ε.

976 M. Habibi, B. Yousefi

Therefore,Tmk

1 T nk2 (x + Sk(zr,s) ∈ B(zr,s, 2ε) ⊂ Vr,s

and soT p

1 T q2 (xp,q)) ∈ T r

1 T s2 (Vr,s) = V.

Now since xp,q ∈ U , thus T p1 T q

2 (U) ∩ V �= φ. This completes the proof.

References

[1] J. Bes, Three Problem on Hypercyclic Operators, PhD Thesis, Kent StateUniversity (1998).

[2] J. Bes, A. Peris, Hereditarily hypercyclic operators, J. Func. Anal., 167,No. 1 (1999), 94-112.

[3] P.S. Bourdon, Orbit of hyponormal operators, Mich. Math. Journal, 44(1997), 345-353.

[4] P.S. Bourdon, J.H. Shapiro, Cyclic phenomena for composition operators,Memoirs of Amer. Math. Soc., 125 (1997).

[5] P.S. Bourdon, J.H. Shapiro, Hypercyclic operators that commute with theBergman backward shift, Trans. Amer. Math. Soc., 352, No. 11 (2000),5293-5316.

[6] G. Costakis, M. Sambarino, Toplogically mixing hypercyclic operators,Procedings of the Amer. Math. Soc., 132, No. 2 (2003), 385-389.

[7] R.M. Gethner, J.H. Shapiro, Universal vectors for operators on spaces ofholomorphic functions, Proc. Amer. Math. Soc., 100 (1987), 281-288.

[8] C. Kitai, Invariant Closed Sets for Linear Operators, Thesis, University ofToronto (1982).

[9] H.N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc., 347(1995), 993-2004.

[10] B. Yousefi, H. Rezaei, Hypercyclic property of weighted composition oper-ators, Proc. Amer. Math. Soc., 135, No. 10 (2007), 3263-3271.