CC BY
43Probl. Anal. Issues Anal. Vol. 9 (27), No3, 2020, pp. 83-98
83
DOI: 10.15393/j3.art.2020.8750
UDC 517.98
C-C. Chen, S. M. Tabatabaie
CHAOTIC AND HYPERCYCLIC OPERATORS ON SOLID BANACH FUNCTION SPACES
Abstract. In this paper, we study hypercyclicity on solid Banach function spaces, and give the characterization for weighted translation operators to be hypercyclic in terms of weight and aperiodic functions. Some sufficient and necessary conditions for these operators to be chaotic are obtained as well.
Key words: hypercyclic operator, chaotic operator, weighted translation operator, solid Banach function space, aperiodic function
2010 Mathematical Subject Classification: 47A16
1. Introduction and Preliminaries. Hypercyclicity, as an active research topic in mathematics, arises from the invariant closed subset problem, which is one of the important and significant problems in analysis. Besides, it is related to some notions of topological dynamics, such as topological transitivity, topological mixing, linear chaos, and so on. Indeed, fruitful results and theories appeared during the last four decades. We refer to these two classic books [2], [10] as monographs on this topic.
Among the works in this direction, H. Salas in [14] gave a concrete example on £p(Z) by characterizing hypercyclic weighted shifts, which is very important to motivate researchers to demonstrate some deep results and construct examples in various cases. Inspired by H. Salas' work, the authors of [6], [7] gave some sufficient and necessary conditions for weighted translation operators to be hypercyclic on the Lebesgue space in context of homogeneous spaces and locally compact groups.
Recently, the focus was on studying hypercyclicity of operators on other special function spaces, such as Orlicz spaces; see [8], [9]. In this note, we consider hypercyclicity on more general spaces, namely, Banach function spaces, which are Banach spaces of measurable functions. Indeed, we characterize hypercyclic weighted translation operators on such spaces.
© Petrozavodsk State University, 2020
Also, we give some sufficient and necessary conditions for these operators to be chaotic, and present some application in Morrey spaces.
First, we recall some preliminaries and definitions of Banach function spaces, and refer to [3] or another classic monograph. Let X be a topological space. Denote the set of all Borel measurable complex-valued functions on X by Mo(X).
Definition 1. Let X be a topological space and F be a linear subspace of M o(X). If F equipped with a given norm || • ||f is a Banach space, we say that F is a Banach function space on X. A Banach function space (F,|| • ||f) on X is called solid if for each f E F and g E M0(X), the inequality |g| ^ |f | implies g eF and ||g||F ^ ||f ||f.
Definition 2. Let F be a Banach function space on a topological space X, and a : X — X be a Borel measurable bijection, whose inverse a-1 is also Borel measurable. We say that F is a-invariant if f o a±1 E F and ||f o a±1 ||f = ||f ||f for each f eF .
Example 1. Let G be a locally compact group, a be a fixed element in G, and $ be a Young function. Let (L^(G),|| • ||$) be an Orlicz space with respect to a left Haar measure on G. Define the function aa : G — G by
aa(x) := ax, (x E G).
Then L®(G) is an aa-invariant solid Banach function space on G. In the following, we recall the definition of hypercyclicity.
Definition 3. Let X be a Banach space. A bounded linear operator T : X —y X is called hypercyclic if there exists an element x E X, such that the orbit {x,Tx,... ,Tnx,...} is dense in X; here Tn denotes the n-th iterate of T. In this case, x is called a hypercyclic vector.
It is well-known (cf. [10]) that topological transitivity and hypercyclicity are equivalent on X. An operator T is said to be topologically transitive if for any pair of non-empty open sets U,V in X, there exists n E N, such that Tn(U) fl V = 0. Besides, it should be noted that a Banach space admits a hypercyclic operator if, and only if, it is separable and infinite-dimensional [1], [4]. So, in this paper, we assume that Banach spaces are separable and infinite-dimensional. Next, we introduce our setting.
Definition 4. Let X be a topological space and a : X — X be a Borel measurable bijection, such that its inverse is also Borel measurable. Let F be an a-invariant solid Banach function space on X and let
w : X ^ (0, to) be a bounded Borel measurable function (called a weight). Then, the corresponding (generated) weighted translation operator Ta,w on F is defined by
Ta,w : F^F, Ta,w f := w • (f o a)
for all f e F.
Easily, one can see that Ta,w is well-defined, and by some calculation, for each n e N and f e F, we have
n— 1
T«nwf =( n w o aj) • (f o an) j=o
where an means n-fold combination of a. If H is also bounded, then Ta,w is invertible and we denote its inverse by Sa,w := T—1w.
In order to pursue our goal, one more concept is required.
Definition 5. Let X be a topological space, and a : X ^ X be a Borel measurable bijection, such that its inverse is also Borel measurable. Then a is called aperiodic if for each compact subset K of X there exists a constant N > 0, such that for each n ^ N, K n a±n(K) = 0.
Example 2. Aperiodicity of elements of a locally compact group G was defined originally in [11]. In fact, an element a e G is called compact in [11]) if the closed subgroup of G generated by a is compact, and any non-compact element of G is called aperiodic in [7]. Let G be a second countable locally compact group; fix an element a e G. As in Example 1, define the function aa : G ^ G by aa(x) := ax for all x e G. Then, by the characterization given in [7, Lemma 2.1], aa is an aperiodic function if, and only if, a is an aperiodic element of G. We note (see [7]) that in many familiar non-discrete groups, including the additive group Rd, the Heisenberg group and the affine group, all elements except the identity are aperiodic. Therefore, one can construct many aperiodic functions a on various groups.
2. Main Results. In this section, we will provide and demonstrate the main results. We denote by xe the characteristic function of E. To remind ourselves of all the required conditions and properties on the Banach function space F, we collect them below.
Definition 6. Let X be a topological space, F be a Banach function space on X, and a be a Borel measurable bijection from X onto X, such
that a-1 is also Borel measurable. Let Fbc be the set of all bounded compactly supported functions in F. We say that F satisfies condition Qa if
1) F is solid and a-invariant;
2) for each compact set E C X, we have \e G F;
3) Fbc is dense in F.
Example 3. Let G be a locally compact group, a be a fixed element in G and aa be the mapping given in Example 1. Let $ be a A2-regular Young function. Then the Orlicz space L^(G) satisfies condition Qaa. We recall that a Young function $ is called A2-regular if there are some constants c> 0 and x0 ^ 0, such that $(2x) ^ c $(x) for all x ^ x0. Now we are ready to give the main results of this paper.
Theorem 1. Let X be a topological space and a be an aperiodic function on X. Let w be a given weight on X, and w be bounded. Let F be a Banach function space on X satisfying condition Qa. Then the following properties are equivalent.
(i) Ta,w is hypercyclic on F.
(ii) For each compact subset K of X, there are a sequence of Borel subsets (Ek)£=1 of K, and a strictly increasing sequence of natural numbers (nk)fc=1, such that \\XK\Ek ||f = 0 and
nk -1 nk
lim sup I I ((w o aj)(x)) = lim sup I |(w o a-j)(x) = 0.
k^ xeEk xeEk jJl
Proof. (i) ^ (ii). Suppose that Ta,w is hypercyclic, and K C X is compact. Then, by the condition (2) of Definition 6, we have \K G F. Since a is an aperiodic function, there exists a constant M > 0, such that for each n ^ M,
K n a±n(K) = 0. (1)
Since Ta, w is hypercyclic, the set of all hypercyclic vectors of Ta, w is dense in F. So, for each k G N, there is a hypercyclic vector fk in F, such that
\\fk - Xk\\f ^ 4k. (2)
Due to the fact that fk is a hypercyclic vector, there is a natural number nk, such that
\\Tankwfk - Xk\\f < ^. (3)
We can suppose that n1 > M, and the sequence (nk)(j=1 is strictly increasing. Put Ak := {x e K : lfk(x) — 1| ^ 2k}. Then, for each x e K \ Ak,
we have
1 -lfk (x)| ^ lfk (x) - 1| < -,
and so, 0 < 1 — 2k < lfk(x)|. Thus,
IIXAfc fk — XAk If ^ \\fk — XkIf ^ 4k (4
which is implied by the fact that fk — xk e F, the solidity of F, and
lXAk fk — XAk1 = lXAk fk — XAk Xk 1 = XAk lfk — Xk 1 ^ fk — Xk |.
Moreover,
which entails
2k
XAk
2k
XAk
f
^ lXAk fk - XAk I
^ \\XAk fk - XAk I
T.
By (4) and (5), we have \\xAk If ^ 2k. Now, we put Bk := {x e X \ K : lfk(x)| ^ 2k}. Then, for each x e X \ (K U Bk), we have lfk(x)| < 2k. Also, since Bk n K = 0, we have
lXBkfk1 = lXBkfk — XBknK1 = lXBk • (fk — Xk)| =
= XBk lfk — Xk 1 < lfk — Xk I
which implies xb fk e F by solidity of F. Therefore,
\\XBk fk\f ^ II fk — Xk\f ^ ^. (6)
Besides, by the definition of Bk, we have 0 ^ 2k XBk ^ XBk lfk |. So,
2k
XBk
f
^ \\XBkfk\\f.
Using (6) and (7), we have \\xBk \\F ^ 2k. On the other hand, for all k e N,
1
1
1
& > Rwfk - xk||
f
nk-1
n
3=0
nk-1
n
3=0
JJ^ w o a3 o a nk
n w o j • (fk o ank) - xk - (Xk o a-nk) (nw o aj fk - (xk o a-nk)
j=1
F
For each k G N, put
nk-1
3=0
Ck :={ x G K : |( n (w o a3 )(x)) • (fk o ank) (x) - l| ^ 2k}
Then, by (3) and the solidity of F, we have
l
4k >
nk-1
>
3=0
nk-1
XCk[ II
3=0
Ilw o a3) • (fk o ank) - xk
>
F
w o a3) • (fk o oink) - XCk
>
F
2k
Xck
F
2k \\xck\\f
which says \\xck\\F < 2k. Next, set
D
nk
n(w j=1
k x G K : (w o a-j)(x) • |fk(x)| ^ -
l
2k
If x G K, then clearly, XDk(x).xK (a-nk(x)) = 0. Let x G K. By ape-riodicity of a, one has a-nk (K) n K = 0, and so a-nk (x) G K. Hence XDk (x).XK (a-nk (x)) = 0. Together with (3), this imply
l
4k >
nk
(n
j=1
w o a 3 )fk - (xk o a nk
f
( nk
XDk[Ylw o a j=1
-3
>
f
>
l
2
XDk
l
= 2k llxdk\\f
which implies \\xDk\\f < 2k.
l
Next, we show that for each x E K \ (Ck U a nk(Bk)),
fri1 w ◦ a (x))"' s ^-an;ix)i < ^ __L
j=0 1 2k 1 2k 2k 1
If x e K\ (Ck U a—nk(Bk)), then x e K, x / Ck and ank(x) e Bk. Since ank (K) P| K = 0 and x e K, we have ank (x) e X \ K. Applying the definition of Bk, we have the following estimate:
|(fk o ank)(x)| = lfk (ank(x))| < 2k,
and so,
l(fk ◦ ank)(x)| < 2k
1 - -1 1 - -1'
— 2 k — 2 k
Moreover, since x e Ck, by the definition of Ck, one has
nk -1
(,„naj )(x)\ l(f, D ank )(x)| <
2k
j=0 which says
nk -1
1 -( n (w ◦ aj)(x)) l(fk ◦ ank)(x)| <
1
n(w ◦ aj )(x)) * 4 fk ◦ a '(x)l. (10)
j=o 7 1 -
Now, (8) can be immediately deduced from (9) and (10) On the other hand, we are going to show that
nk 1 1
j[(w o a-j )(x) < < T, x e K \ (Dk U Ak). (11)
With this aim, assume that x e K \ (Dk (J Ak). Then lfk (x)| > 1 — 2k by the definition of Ak. Hence,
2k < . (12)
lfk(x)l ^ 1 - 2k'
Also, by the definition of Dk, we have
nk l
n(w o a-3)(x)) |fk(x)| < 2k, 3=1
and so
m 1
n(w o "" )(x) < fwr (13)
Hence, one can obtain (11) by (12) and (13). Now, put Ek =K \ (Ak U a-nk (Bk) U Ck U Dk)
We claim that
\\ \\ 4 \\XK\Ek\\f < 2k.
Indeed, since
xK\Ek ^ xAk U a-nk (B) U Ck U Dk ,
we have
\\XK\Ek\\f ^ \\xAk\\f + \\X a nk (Bk)||f + \\XCk\\ + \\XDk\\ f
4
= \\XAk\\
+ \\XBk\\ + \\XCk\\ f + ||xDk\\f < 2k . Moreover, using (8) and (11), for each x G Ek, we have
nk-1 ^ -1 l nk l
II(w ◦a)(x)) ^ and n(w ◦a3)(x) ^ 2k- 1
3=0 3 = 1
Therefore, condition (ii) follows.
(ii) ^ (i). Assume that condition (ii) holds. We are going to show that Taw is topologically transitive. For this, let U and V be non-empty open subsets of F. Since Fbc is dense in F, there are functions f,g G Fbc, such that f G U and g G V. Put
K := supp(f ) ^J supp(g).
Then K is compact. Since a is aperiodic, there is a constant M > 0 such that for each n ^ M, Kna±n(K) = 0. Besides, for the set K, there are a
sequence of Borel subsets (Ekof K, and a strictly increasing sequence of natural numbers (nksatisfying condition (ii). Here we may assume nk > M for each k G N. Thus,
t::w (fxEk )
f
nk
nk-1 ^
JJ (W ◦ Ol3 ^ ■ (fXEk ) ◦ a j=0
nk
f
II(w O a-3)) ■ (fXEk)
3=1
nk
<
f
F ■ sup J^[(w o a 3 )(x) xeEk 3=1
for each k E N. Hence, by condition (ii),
Jim \\Tnkw(fXEk)|
f
0.
:i4)
lim \\sakw (gxEk )\
fc^œ
f
0.
:i5)
Similarly, For each k E N, let
vk := fXEk + Sa>;w (9XEk). Then, vk E F and, for each k E N, we have
\\vk — f If ^ If - fXEk If + \\S2kw (gXEk )\\f = = \\f • XK\Ek If + \\sakw (9XEk )\\F ^ ^ If \\sup \\XK\Ek\\f + \\S'2kw (9XEk )\F .
So, lim vk = f in F. Also, by (14), we have lim Tnkwvk = g in F, which
k—y^o k—y^o '
is implied by the inequalities Igl (1 — XEk) ^ Igl XK\Ek and
WTnkwVk - gW
f
WTnkw(fXEk) + gXEk - gWF ^
< WTnkw(fXEk)Wf + WgWsup WxK\EkWf.
Therefore, U H T nk(V) = 0. This implies that Ta>w is topologically transitive, and the proof is complete. □
In the following, we give some sufficient and necessary conditions for a weighted translation to be chaotic. First we recall the definition of chaos.
Definition 7. Let X be a Banach space, and T be a bounded linear operator on X .A vector x E X is called a periodic element of T if there
exists a constant N e N, such that TNx = x. The set of all periodic elements of T is denoted by P(T). An operator T is called chaotic if it is topologically transitive and P (T) is dense in X.
Theorem 2. Let X be a topological space, a be a Borel measurable bijection from X onto X, such that a-1 is also Borel measurable, and let w be a weight on X. Let F be a Banach function space on X satisfying condition Qa. Assume that for each compact subset E C X, there exists a sequence of Borel subsets (Ekof K, such that lim \\xK\Ek \\F = 0
k^x
and
x lnk
k £ II (n
w ° a ^ ■ XEk
l=1 j=1
+
t
x Ink — 1
11 ( n w ° aj l=1 j=0
1
XEk
F
0,
for some strictly increasing sequence (nk)X=1 C N. Then Ta>w is chaotic on F.
Proof. First we show that Ta,w is topologically transitive. Let U and V be non-empty open subsets of F. We prove that there exists a number n e N, such that Tn w(U) fl V = 0. Since Fbc is dense in F, there are functions f, g, such that f e U fFbc and g e V flFbc. Put K := supp(f) U supp(g). So, K is compact. Then we can pick an increasing sequence (nk) C N and a sequence of subsets (Ek) of K satisfying the assumptions of the statement. Due to the inequality
\\Tnw (fXEk )\\
f
nk — 1
n w ° aj) ■ (fXEk) ° ank j=0
nk
Uw ° a ^ ■ (fXEk) j=1
<
sup
t
f
nk nnk
j=1
w ° a ^ ■ XEk
f
one has lim Tnkw(fXEk) = 0 in F. Similar to the above argument, we
can see that lim S'nkw (gXEk) = 0 in F. (Note that in this theorem, it is
not necessary to assume that 1 is bounded. This is because here f and
J w J
g are compactly supported.) Therefore, together with lim ||xK\Ek||F = 0,
k^x
we have
,lim (snkw(gxEk) + fxEk) = f and lim T^wis^w(gxEk) + fxEk) = g.
k^-x k^x
This implies that Tnkw(U) H V = 0 for some k G N, which tells us that Ta>w is topologically transitive.
Next, we prove that P(Ta,w) is dense in F. Recall that f G U H Fbc. For each k G N, let
x x
Vk := fxEk + £ Tlankw (fxEk ) + £ Sl:kw (fxEk ). l=1 l=1
x
Note that the series T^w (fxEk ) is convergent in F because it is abso-1=1 '
lutely convergent. Indeed,
lnk
< CO.
.. . J k F
1=1 1=1 j=0
£ \Kkw (fXEk )\\f ^ 11/\\suP £ MU w ◦ aj • XEk
Similarly, ^ S^W(fXEk) is convergent in F, too. Using lim ||xk\eJ|f = 0,
l=l ' k^-x
we have lim vk = f in F. In addition, a simple calculation gives Tnkwvk = vk.
k^-x '
This implies that U HP(Ta,w) = 0. Therefore, P(Ta,w) is dense in F. □
Example 4. Let 1 ^ q ^ p < ro. Let f G Lqoc (Rn). Then its Morrey norm is defined by
r i
sup \B(x, r)|p - U If (y)\qdy) q , (16)
Clnyin ^ V./Di'„. „.i /
MP
(x,r)eRnx(0,x) yJB(x,r)
where B(x,r) is the open ball centered by x with radius r. In this case, the set of all Lqoc(Rn)-functions f with ||f ||m? < ro is denoted by Mp(Rn), and (Mp(Rn),|| ■ ||mp) is called a Morrey space. Morrey spaces are generalizations of the usual Lebesgue spaces. In fact, for each 1 ^ p < ro, we have Mp(Rn) = Lp(Rn). Morrey spaces were introduced by J. Peetre in [13], which was originally motivated by [12]. For more details and references, see [15-17]. For each 1 ^ q < p < ro, the Morrey space
Mp(Rn) is not separable. However, if we consider the Mp(Rn)-closure of Lco(Rn)nMp(Rn), denoted by Mp(Rn), then the Banach space Mp(Rn) is separable and infinite-dimensional, where LC°(Rn) is the set of all functions in Lc(Rn) with compact support. Also, for each 1 ^ q < p < to, the set Mp(Rn) \ Lp(Rn) is spaceable and large enough (see [5, Theorem 2.2]). Note that a subset S of a topological vector space X is called spaceable if S U {0} contains a closed infinite-dimensional subspace of X.
As we mentioned in Example 1, for each non-zero element a G Rn, the function aa defined by aa(x) := x — a for all x G Rn is an aperiodic function. Therefore, by the above facts, the Banach space Mp(Rn) satisfies the condition Qaa, because the Lebesgue measure on Rn is additive-invariant.
Applying Theorem 2 and Example 4, we can conclude the following result for Morrey spaces.
Corollary 1. Let w be a weight on Rn, and let a G Rn be a non-zero element. Assume that for each compact subset K C Rn, there exists a sequence of Borel subsets (Ek)C=1 of K, such that lim \\xK\Ek Hmp = 0
fc^c q
and
lim ( }
k—>oo V
1=1
lnk j=1
XEk
MP
+
œ lnk — 1
+ n + ja)
l=1 j=0
1
XEk
MP
for some strictly increasing sequence (nk)^=1 Ç N. Then Ta on Mp(Rn).
is chaotic
We end this paper by giving a necessary condition for chaos under one more assumption. For each f G F, let a(f) := {x G X : f(x) = 0}. In Theorem 3, we assume that there exists a constant p > 0, such that If + 9\\F = If IIF + IMIF whenever f,g G F and a(f ) n a(g) = 0. In fact, this property appears in many familiar Banach spaces.
Theorem 3. Let X be a topological space, a be an aperiodic function from X onto X, and let w be a weight on X. Let F be a Banach function space on X satisfying condition Qa and the additional condition above. Assume that Ta,w is chaotic on F. Then, for each compact subset K Ç X, there exists a sequence of Borel subsets (Ek)£=1 of K, such that
0
lim \\XK\Ek \\f = 0 and
k—y^o
lim ( }
fc—to V
l=i
lnk
n
j=i
w 0 a j I • XEk
+
f
E
1=1
lnk -1
w 0 et
j=0
-1
• XEk
for some strictly increasing sequence (nk)C=1 C N.
Proof. Suppose that Ta>w is chaotic and K C X is compact. Then, by the condition Qa, we have \K G F. Since a is an aperiodic function, there exists a constant M > 0, such that for each n ^ M,
K n a±n(K) = 0.
:i7)
For each k E N, by the density of periodic elements of Ta>w, there exists fk E P(Ta,w), such that Tnkwfk = fk = Snkwfk and
fk - XK\\f < 4k
:is)
where we may assume that n1 > M, and (nk)C=1 is strictly increasing. Put Ak := {x G K : lfk(x) — 1| ^ 2k}. Then, as in the proof of Theorem 1, we have
1
1
lfk(x)| > 1 on K \ Ak, and \\xAk\\f <
2k
2k'
Let Ek = K \ Ak. Then
\\XK\Ek\\f = \\XAk\\f < 2k. On the other hand, by the fact K n a±nk (K) = 0, we observe
XU=i(a-lnk (K)Ual"k (K))fk
XU£i(.a-l"k (K)Ual"k (K))Jk — XU'¡=1 (a-lnk (K)Ual"k (K))XK
Jl=1
XUi= 1(a-lnk (K )Ualnk (K ))\fk - XK \ ^ \fk - XK\.
< \\fk - Xk\\f.
It follows that
XUi=1(a-lnk (K)Ualnk (K))
f
P
0
Hence, by a-invariance, and the disjointness of arnk (K) and asnk (K) for r = s, we get
4kp
>
XUi=i(a-lnk (K)Ualnk (K))
P
T
£ Xa-lnk (K) fk +£ Xc
Ca-lnk (K)Jk + J Xalnk (K), l=1 l=1 xx
ia-lnk (K)Jk\\PF + £ \\Xalnk (K) fk \\F = l=1 l=1
\\Xa
£ \\xk(fk ◦ a-lnk)\f + £ \\xk(fk ◦ alnk)\\f =
1=1
1=1
oo
£ \\xk((Tjnw fk) ◦ a-lnk)\f + £ \\xk((SanWfk) ◦ alnk)
1=1 1=1
x Ink — 1
£ \\xk( n w 0 aj 0 a—lnk) ■ (fk 0 alnk 0 a—lnk) l=1 j=0
x Ink
£ \\xk ( n w o a-j o alnk) — 1 ■ (fk o a-lnk o alnk)
+
l=i
j=i
This implies that
4kp
>
£
l=i
> 11 - 2k
lnk p x Ink — 1
n w 0 a—^ ■ fkXEk + £\\( n w 0 aj
j=1 l=i j=0
lnk x lnk — 1
i
■ fk XEk
>
T
\
l=1 j=1
w0a j ¡■XEk
l=1
I \\ y JJ^ w0aj j=0
-1
■ X Ek
Therefore, the condition given in the conclusion follows. □
Acknowledgment. We would like to thank the referee for carefully reading our manuscript and for his/her helpful comments.
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Received July 16, 2020. In revised form, November 02, 2020. Accepted November 03, 2020. Published online November 15, 2020.
C-C. Chen
Department of Mathematics Education, National Taichung University of Education, No.140, Minsheng Rd., West Dist., Taichung 40306, Taiwan E-mail: chungchuan@mail.ntcu.edu.tw
S. M. Tabatabaie
Department of Mathematics, Faculty of Science, University of Qom, Alghadir Blvd, Qom 3716146611, Iran E-mail: sm.tabatabaie@qom.ac.ir
