1. Far East Journal of Mathematical Sciences (FJMS)
Volume 47, Number 2, 2010, Pages 219-223
This paper is available online at http://pphmj.com/journals/fjms.htm
© 2010 Pushpa Publishing House
:tionClassificajectSubsMathematic2010 47A16.
Keywords and phrases: hypercyclic vector, n-tuple, hypercyclicity criterion, Hilbert space.
Received August 20, 2010
n-TUPLES AND EPSILON HYPERCYCLICITY
MEZBAN HABIBI and FATEMEH SAFARI
Islamic Azad University, Branch of Dehdasht
P.O. Box 7571763111, Dehdasht
Iran
e-mail: m.habibi@farsedu.ir
safari6111@yahoo.com
Abstract
In this paper, we study the epsilon hypercyclicity on n-tuple. In fact, this
paper is a generalization of Theorem 1.4 of [6] for n-tuples.
1. Introduction
A nice criterion, namely the Hypercyclicity Criterion is used in the proof of our
main theorem. It was developed independently by Kitai [7], and Gethner and Shapiro
[5]. This criterion has used to show that hypercyclic operators arise within the class
of composition operators [4], weighted shifts [9], and adjoints of subnormal and
hyponormal operators [3], and Hereditarily operators [2]. The formulation of the
Hypercyclicity Criterion in the following theorem was given by Bes Ph.D. thesis [1].
Many mathematicians generalized these properties to n-tuples. The first example of a
hypercyclic operator on a Hilbert space was constructed by Rolewicz in [8]. For
more information please, see, ([1]-[10]).
Definition 1.1. Let nTTT ...,,, 21 be commutative bounded linear operators on a
Banach space .X Then, for n-tuple ( ),...,,, 21 nTTT=T put
{ }.0...,,,: 2121
21 ≥=Γ n
m
n
mm
mmmTTT n

2. MEZBAN HABIBI and FATEMEH SAFARI220
For ,X∈x the orbit of x under T is the set ( ) ( ){ },:, Γ∈= SxSxOrb T that is,
( ) { ( ) }.0...,,,:, 2121
21 ≥= n
m
n
mm
mmmxTTTxOrb nT
The vector x is called hypercyclic vector for T and n-tuple T is called hypercyclic
n-tuple, if the set ( )xOrb ,T is dense in ,X that is,
( ) { ( ) } .0...,,,:, 2121
21 XT =≥= n
m
n
mm
mmmxTTTxOrb n
Definition 1.2. Let nTTT ...,,, 21 be commutative bounded linear operators on a
Banach space ,X ε be a number in ( )1,0 and x be a vector of .X Then the vector x
is called ε-hypercyclic vector for n-tuple ( )nTTT ...,,, 21=T and the n-tuple T is
called ε-hypercyclic n-tuple, if for every non zero vector ,X∈y there exist some
integers nmmm ...,,, 21 such that
.21
21 yyxTTT nm
n
mm
ε<−
Note that, all operators in this paper are commutative operator on a Banach
space .X
2. Main Results
Theorem 2.1 (The Hypercyclicity Criterion). Let X be a separable Banach
space and ( )nTTT ...,,, 21=T be an n-tuple of continuous linear mappings on .X
If there exist two dense subsets Y and Z in ,X and n strictly increasing sequences
{ } { } { }njjj mmm ,2,1, ...,,, such that:
(1) 0,2,1,
21 →njjj m
n
mm
TTT on Y as ,∞→j
(2) there exists function { }X→ZS j : such that for every ,0, →∈ zSZz j
and ,,2,1,
21 zzSTTT j
m
n
mm njjj
→
then T is a hypercyclic n-tuple.
Theorem 2.2. Let X be a separable Hilbert space and ( )nTTT ...,,, 21=T be
an n-tuple of commutative bounded linear operators on a Hilbert space .X If, for
every ,0>ε the n-tuple T is ε-hypercyclic, then T is a hypercyclic n-tuple.

3. n-TUPLES AND EPSILON HYPERCYCLICITY 221
Proof. Note that, if T is a hypercyclic n-tuple, then ( ) ,φ=σ ∗
Tp ( =∗
T
( )),...,,, 21
∗∗∗
nTTT also all spaces that admitted some hypercyclic operator, are
infinite dimensional spaces, so we can assume that X is infinite dimensional space.
Suppose that U and V are subset of .X Give ,, VU ∈∈ vu two nonzero element
and 0>δ so large that ( ) U⊂δ,uB and ( ) V⊂δ,vB so that { }., vuMax<δ
Take x such that x is an ε-hypercyclic for T with property
{ }
,
,6 vuMax
δ
<ε
then there exist 0,0,20,1 ...,,, nmmm such that
δ<ε<− uuxTTT nm
n
mm 0,0,20,1
21
thus we have
.21
21 U∈xTTT nm
n
mm
Suppose on the contrary that there are only finitely many such integers
....,,,
,...,,,
,...,,,
,,2,1
2,2,22,1
1,1,21,1
tntt
n
n
mmm
mmm
mmm
As above, for each X∈′u with
,
3
2δ
<−′ uu
there exist integers
( ) ( ) ( )umumum n ′′′ ...,,, 21
which satisfy
( ) ( ) ( )
.
3
221
21
δ
<ε′ε≤′−
′′′
uuuxTTT um
n
umum n
Since
( ) ( ) ( ) ( ) ( ) ( )
,2121
2121 δ<′−+−≤′−
′′′′′′
uuuxTTTuxTTT um
n
umumum
n
umum nn
we have
( ) { }knkkk mmmum ,,2,1 ...,,,∈′

4. MEZBAN HABIBI and FATEMEH SAFARI222
for tk ...,,2,1= and the ball ⎟
⎠
⎞
⎜
⎝
⎛ δ
3
2
,uB is covered by finite number balls
,
3
,1,1,21,1
21 ⎟
⎠
⎞
⎜
⎝
⎛ δ
xTTTB nm
n
mm
,
3
,2,2,22,1
21 ⎟
⎠
⎞
⎜
⎝
⎛ δ
xTTTB nm
n
mm
.
3
,,,2,1
21 ⎟
⎠
⎞
⎜
⎝
⎛ δ
xTTTB tntt m
n
mm
Thus, in an infinite dimensional space, this is impossible. So there are infinitely
many integers as nmmm ...,,, 21 with
.21
21 δ<− uxTTT nm
n
mm
Then there exist 0,, iki mm > for tk ...,,2,1= and ni ...,,2,1= such that
.21
21 V∈xTTT nm
n
mm
Thus
xTTTTTT nnnn m
n
mmmm
n
mmmm 0,0,20,10,,0,22,20,11,1
2121
−−−
belongs to
( ),0,,0,22,20,11,1
21 UTTTV nnn mm
n
mmmm −−−
∩
that is,
.0,,0,22,20,11,1,2,21,1
2121
nnnnn mm
n
mmmmm
n
mm
TTTVxTTT
−−−
∈ ∩
Here it can be concluded that T is hypercyclic n-tuple.
Corollary 2.3. Let ( )21, TT=T be a pair of commutative bounded linear
operators 21, TT on a Hilbert space .X If, for all ,0>ε the pair T is
ε-hypercyclic, then T is a hypercyclic pair.
Proof. In Theorem 2.2, take ,2=m the proof is similarly.

5. n-TUPLES AND EPSILON HYPERCYCLICITY 223
Acknowledgement
This research was partially supported by a grant from Research Council of
Islamic Azad University, branch of dehdasht, so the authors gratefully acknowledge
this support.
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