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Vladikavkaz Mathematical Journal 2020, Volume 22, Issue 3, P. 112-123
УДК 517.98
DOI 10.46698/p4238-0191-2122-t
BOUNDED COMPOSITION OPERATORS ON WEIGHTED FUNCTION SPACES IN THE UNIT DISK
S. Hua1, L. H. Khoi1 and P. T. Tien2 3
1 School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371; 2 University of Science, Vietnam National University, 334 Nguyen Trai St., Hanoi, Vietnam; 3 TIMAS, Thang Long University, Nghiem Xuan Yem, Hoang Mai, Hanoi, Vietnam E-mail: M170129@e.ntu.edu.sg; lhkhoi@ntu.edu.sg, phamtien@vnu.edu.vn
Dedicated to Professor Yu. F. Korobeinik on the occasion of his 90th birthday
Abstract. We introduce a general class of weighted spaces H (fi) of holomorphic functions in the unit disk D, which contains several classical spaces, such as Hardy space, Bergman space, Dirichlet space. We characterize boundedness of composition operators Cv induced by affine and monomial symbols p on these spaces H (fi). We also establish a sufficient condition under which the operator Cv induced by the symbol p with relatively compact image p(D) in D is bounded on H (fi). Note that in the setting of H (fi), the characterizations of boundedness of composition operators Cv depend closely not only on functional properties of the symbols p but also on the behavior of the weight sequence fi.
Key words: composition operator, weighted space, weight sequence, holomorphic function, unit disk. Mathematical Subject Classification (2010): 30D15, 47B37.
For citation: Hua, S., Khoi, L. H. and Tien, P. T. Bounded Composition Operators on Weighted Function Spaces in the Unit Disk, Vladikavkaz Math. J., 2020, vol. 22, no. 3, pp. 112-123. DOI: 10.46698/p4238-0191-2122-t.
1. Introduction
Let H be a space of holomorphic functions on some domain G of the complex plane C. For a holomorphic self-map ^ of G, the composition operator C^ is defined as
Cf = f ◦ <p, f e H.
Researchers are interested in the relation between the operator properties of Cv and the functional properties of the symbol Questions about boundedness, compactness, compact difference, essential norm, cyclicity, complex symmetry and so on, have been considered. These topics are interesting and have received great attention during the past several decades. We refer the reader to the excellent monographs [1-3] and references therein for more detailed information.
2020 Hua, S., Khoi, L. H. and Tien, P. T.
Many studies have been done on composition operators on various different spaces of holomorphic functions on the unit disk or the unit ball, such as Hardy spaces, Bergman spaces, Dirichlet spaces, spaces of bounded holomorphic function and weighted Banach spaces with sup-norm (see, e.g., [4-9]).
Recently, in [10] a general class of weighted Hardy spaces of entire functions, which contains the Fock and Fock-type spaces as well as several other well-known spaces, has been considered and bounded composition operators on these spaces have been studied. It is interesting that the characterization for boundedness of such operators depends closely on the behavior of the weight sequence.
The aim of this paper is to introduce a general class of weighted spaces of holomorphic functions in the unit disk and study two special forms of bounded composition operators on these spaces. This newly introduced class of spaces includes classical Hardy, Bergman and Dirichlet spaces.
So far as we know, this class of spaces has not been treated before. We hope that our approach may inspire the reader to investigate further properties and obtain more general results in the future.
The structure of the paper is as follows. In section 2, the class of weighted spaces H(5) of holomorphic functions in the unit disk is performed and a short but essential comparison with the case H(5, E) of entire functions is discussed. Section 3 deals with boundedness of composition operators induced by affine symbols while in Section 4, the monomial symbols are of our interest.
2. Preliminaries
In 1974 Shields [11] introduced the following weighted spaces. Let a domain G contain the origin and 5 = {5n}£=0 be a sequence of positive numbers. The weighted space H(5) is defined as follows:
/ \ 1/2
f (z) = £ anzn € O(G) : ||f || = PP K|252 < o
n=0 \n=0 J
where O(G) is the space of all holomorphic functions on G. This space H(5) is an inner product space with
(f>9) = Yhanbn^
n=0
for every f (z) = anzn and g(z) = bnzn. Note that H(5) is not necessarily
a complete normed space.
Denote by r the radius of the smallest open disk centered at the origin which contains G. It is well known that H(5) is a Hilbert space if and only if lim infn—TO r. In particular,
in case limn^^ 5r!n = we have a Hilbert space of entire functions H(5, E). It is, however, worth to note that since the case of entire functions H(5, E) is essentially different from the one of holomorphic functions H(5) and moreover, it is already considered in [10], we eliminate those weights for which limn^^ 5rln = oo.
So throughout this paper, we pay our attention to a model case when G is the open unit disk D = {z € C : |z| < 1} and so the condition
1 < liminf 5n/n < o (1)
n—^^o
is supposed to hold.
We note that for fjn = 1, fjn = (n + 1) —2 5 and fjn = (n + 1)2, the space J4?(/3) becomes respectively the classical Hardy space H2(D), Bergman space A2(D), and Dirichlet space D2(D) in the unit disk.
Note also that hn(z) = (n = 0,1,2,...) form an orthonormal basis for the Hilbert
space H(5).
We can easily verify that H(5) is a functional Hilbert space, and hence C^ is bounded on H(5) if and only if it maps H (5) into itself.
It is important to have a remark that in the case of weighted spaces of entire functions H(5,E), bounded composition operators C^ can be induced only by affine functions <^(z) = az + b. However, in the case of weighted spaces H(5) in the unit disk, the situation is much more complicated. It is not true anymore, as we can see in some examples below.
Example 2.1. For the symbol <^(z) = z2, the composition operator C^ is bounded on each of the classical Hardy, Bergman, and Dirichlet spaces; however C^ is unbounded in H(5) in the case 5n = An with A > 1.
< When <^(z) = z2, for every f (z) = anzn € H(5), we have
2 _ V^ u |2 _ II f||2
l|Cf |lH2(ro) = E |an|2 = |f llH2
n=o
II<VI&ck» = £< E Kl2^! = II/II^O).
n=o n=o
II Cf |Id2(d) = E |an|2(2n + 1) < 2 £ |an|2(n + 1) = 2||f |||2(d)• n=o n=o
For /?„ = \n with A > 1, we take a function /(z) = Y.n=i xkz'n G Then
n=0 n=l n=l
which shows that f (z) € H(5). However,
ii^/II2 = E = E ¿¡2 = E ^
n=0 n=1 n=1
which diverges since A > 1. >
Remark 2.2. Replacing <^(z) = z2 by monomials <^(z) = zm with some integer number m > 1, by the same techniques, we can obtain similar results.
3. The Case when <^(z) is Affine
In this section, we study boundedness of composition operators on weighted spaces H(5) induced by affine symbols. This study is inspired by the case of entire functions [10].
Let the symbol <^(z) = az + b (a, b € C). For C^ to be well-defined, <^(z) needs to be a self-map of D.
We have the following simple result.
Lemma 3.1. An affine function <^(z) = az + b (a, b € C) is a self-map of D if and only if either of the following conditions holds:
(i) a = 0 and |b| < 1,
(ii) 0 < |a| < 1 and |b| < 1 - |a|.
< Suppose that <^(z) = az + b is a self-map of D. If a = 0, then |^>(z)| = |b| < 1 and hence we get (i). If a = 0, then ^>(D) is an open disk centered at the point b of radius |a|. Since ^ is a self map of D, ^>(D) C D. In order for the image disk to be fully contained in the unit disk, we have |a| ^ 1 and 1 — |b| ^ |a|, which gives (ii).
Conversely, suppose <^(z) satisfies (i) or (ii). If (i) holds, then |^>(z)| = |b| < 1. In case (ii) holds, |^>(z)| = |az + b| ^ |az| + |b| < |a| + |b| ^ 1 for all z € D. In both cases ^ is a self-map of D. >
From Lemma 3.1, we notice that there are two trivial cases of affine symbols which always induce bounded composition operators. The first case is <^(z) = b with |b| < 1. The second case is <^(z) = az with 0 < |a| ^ 1. In particular, when |a| = 1, this is a unitary operator.
As a result, throughout this section, we are interested only in the symbol <^(z) = az + b satisfying
0 < |a| < 1 - |b| < 1.
(2)
With hn(z) = we get
Thus,
and
(Cvhn,hk) = A.
(;l)akbn~k& n^k] 0, k > n.
(3)
(4)
In the sequel, we consider separately necessary conditions and sufficient conditions for boundedness of C^ on H (S). As it will be seen, several of necessary conditions are "very close" to sufficient ones. We note that some proofs are similar to those of the case of entire functions in [10].
3.1. Necessary conditions. We have following necessary conditions for boundedness of C^ on H (S), which are similar to those appearing in [10] and proved analogously.
Proposition 3.2. Let <^(z) = az + b satisfy (2). Consider three statements:
(i) C^ is a bounded composition operator on H (S).
(ii) {||C^hra||}ra^o is a bounded sequence.
(iii) The following two inequalities hold:
|a| lim sup
k—»00
sup
k + m k
|b|r
t ßk ßk+n
and
un- / f/k + m Ik ßk
|6| lim sup I sup s I |a| -—
m^œ \ k^o IV k / ßk+n
^ 1,
^ 1.
(5)
(6)
Then the implications (i) (ii) (iii) hold. < (i) (ii). Obviously,
||C^h„|| ^ ||C^|op for all n ^ 0, where ||C^||op is the operator norm of C^.
(ii) (iii). By (4) and the Cauchy-Schwarz inequality,
(fc + m)|a|fc|= \(C^hk+m,hk)\ < \\C^hk+m\\
for all k,m ^ 0. From this inequality and boundedness of hn||}n^0, both (5) and (6) follow. >
3.2. Sufficient Conditions. The following lemma is adapted from [10, Lemmas 12 and 13] for the case of entire functions. We sketch the proof for the sake of completeness.
Lemma 3.3. Let ^>(z) = az + b satisfy (2). Suppose that the sequence {||C^h ra||}ra^0 is bounded by some M > 0. Then there exists an integer s € N such that
œ œ
2
£ £ )|2 < fc=Q n=(s+1)fc
< First, let n ^ k ^ 0. By (3) and (4), we have
(1 + |a|)"-*|a|*|(Cyft„,/ît)| = g " ^lal'lal'^lal'l»!"-"^
l=Q n-fc
< E hj < (n - k + 1)M2.
l=Q
This implies that
I (Cvhn, hk)I < fe + x)m2 f n ^ k ^ q
Next, let s € N satisfy (1 + |a|)s|a| > 1. Then putting I = n — (s + 1)k and using the above inequality, we get
œ œ ,2^4 ^ (sk +1+ 1)2
E E
fc=Qn=(s+1)k fc=Q l=Q (1 + |a|) |a|
œ (sk + 1)2 ^ (I + 1)2
Z^ ff1 + I n h s I n 112fc Z^
_{(1 + |a|)s|a|}2kl=Q (1 + |a|)21'
where the first series converges by the choice of s, while the convergence of the second series is obvious. >
Now we are ready to state and prove sufficient conditions for boundedness of C^ on H (S). Proposition 3.4. Let ^>(z) = az + b satisfy (2). Suppose the following condition holds
. fc=i
Then C^ is bounded on H (S).
sup f V k| (C^hn, hfc)|2 ] < TO. (7)
< By (7), there is a number M > 0 such that
n
)|2 < M2 for every n ^ 1.
hk )l2 ^ M2 k=1
Then the sequence {||C^hn||}n^0 is also bounded by M, because
|Chn||2 = £ K^hn,hk)|2. k=0
Hence, by Lemma 3.3, there exists a number s € N such that
to TO
A = E E h nj hfc)|2 < to.
k=0 n=(s+1)k
Take and fix an arbitrary function f (z) = ^TO=0cnhn(z) in H(5). First we compute (Cf hk) in two cases with respect to k. If k ^ 1, then, by (4),
^ (s+1)k-1 TO
(C^fjhk) = ^ Cn(C^hnjhfc) = E Cn(C^hn,hfc) + hk),
n=k n=k n=(s+1)k
which implies that
. / (s + 1)k-1 \ 2 / TO x 2
V n=k / yn=(s+1)k
(s+1)k-1
< sk E |Cn(C^hnjhk)|2 + I E |Cn|2| I E |(C^hnjhk)|2
n=k yn=(s+1)k J yn=(s+1)k
(s + 1)k-1 / TO
< sk E |Cn(C^hnjhk)|2 + ||f||2 I E |(C^hnjhk)|2
n=k yn=(s+1)k
If k = 0, then (Cf jho) = ETO=0 Cn(C^hn,hk) = ETO=(s+1)k c«(C^hn,hk), which implies that
1TO
n=(s+1)k
Next taking summation over k ^ 0 and using the calculations above, we obtain
TO (s+1)k-1 TO / TO
k=1 n=k k=0 yn=(s+1)k
TOn
< sE |cn|^k|(C,hnjhk)|2 + A||f||2 < sM2||f ||2 + A||f||2.
n=1 k=1
From this we conclude that C^ is bounded on
H (£). >
We can modify sufficient condition (7) by a slightly stronger condition.
Corollary 3.5. In Proposition 3.4, condition (7) can be replaced by the following: There exists a function h : {0,1, 2,...} ^ R such that
Ykh2(k)< oo and sup \(k + m\a\k\b\m-^-) < h{k), (8)
m^O A k / J
for every k ^ 0.
< For any n ^ 1, by (4), we have
j^k\(Cvhn,hk)f = < ¿fc^(fc) < oo,
fc=i fc=i ^ ^ ' fc=i
which implies (7). Thus, by Proposition 3.4, C^ is bounded on H (£). > We can further develop condition (8).
Corollary 3.6. In Corollary 3.5, condition (8) can be further replaced by the following inequality
i -
M lim sup sup { r t m )\b\m7r~ Y ) < 1. (9)
fc^œ V m^O IV k / Pfc+m
which is very closed to necessary condition (5) in Proposition 3.2.
< Suppose condition (9) is satisfied. Then there exists r € (0,1) such that
|a| lim sup f sup(P + mVr7r^rl < r < 1.
fc^œ y m^O IA k / Pfc+m J J
For this r, by the properties of lim sup, we can find a constant M > 0 (depending only on r) so that
k + m \ i„ifcmm A I ^ ïi^fc
sup <M \a\ \b\m-f^ f < Mr for a11 k >
m^O IA k / Pfc+m J
Taking h(k) = Mrk, we can easily verify that ^fch2(fc) < to. Thus we obtain condition (8) and so is bounded on H(P). >
Corollary 3.6 shows that the strict version of (5) is a sufficient condition for boundedness of C< on H(P). Using some ideas in the proof of [10, Lemma 18], we prove that the same result is true for the strict version of (6).
Proposition 3.7. Let <^(z) = az + b satisfy (2) such that
k + mY_ifc A
|6| lim sup sup ' M <L (10)
m—^^o \ fc^O L \ k / Pfc+m J /
Then C< is bounded on H (5).
< Similarly to the proof of Corollary 3.6, there exist M > 0 and r € (0,1) such that
supip^Vh&rTr^l < Mrm for all m ^ 0.
fc^O IA k / Pfc+m J
From this and (4), it follows that
\(Cvhk+m,hk)\ = (Kk+km)\a\k\b\m^~ ^Mrm forall (11)
Take and fix an arbitrary polynomial f (z) = ^^o Cnhn(z) € H(P), where only finite many c„ are nonzero. By (4), we get
oo oo oo oo oo
Cf = E= E E Ck+m(C^hfc+m, )hk = E E cfc+m(C^hfc+m, )hk•
k=0 k=0 m=0 m=0 k=0
This and (11) imply that
ic f ii < E
k=Q
m=Q
oo / oo \ 2 oo , ,
2|/^ U U \|2 \ ^ ji-rii ill V^ „m _ M
= E ^Mii/nEr i_r
m=0 \fc=0 / m=0
From this and the density of the set of all polynomials in H (S), the assertion follows. >
Combining Proposition 3.2, Corollary 3.6 and Proposition 3.7, we have the following important summary for boundedness of C^ on H (S). Theorem 3.8. Let <^(z) = az + b satisfy (2).
(1) If the operator C^ is bounded on H (S), then the following two inequalities hold:
(!) lallimsupfsupjmir^j^^l,
fc^TO \m^0 ^ 1 J
(ii) |6| limsup fsup{(fcr)l«lfc^}™) <1-
m^TO \fc^0 L + J /
(2) Conversely, if one of the following conditions hold: (in) |a| lim sup (sup {mi&P^}*) < 1,
(iv) |6| lim sup fsup{(fcr)Hfc^}™) <1,
m^TO \fc^0 L +m J /
then the operator C^ is bounded on H (S).
4. The Case when <^(z) are Monomials
4.1. Monomial symbols. Let the symbol <^(z) = czm (c € C \ {0}, m € N \ {1}). As in the previous section, to define well C^ the symbol should be a self-map of B. We can easily check that it is so if and only if 0 < |c| ^ 1, i.e. c € B \ {0}. Thus throughout this section, such a symbol ^ is supposed to consider.
We can easily prove the following criterion for boundedness of C^ on H (P).
Theorem 4.1. Let c € B\ {0} and m € N \ {1}. The operator C^ induced by the symbol ^>(z) = czm is bounded on H(P) if and only if
|c| Pmra , /io\
sup -———- < oo. (12)
raSN Pra
Moreover, in this case, the above supremum is equal to ||C<||op. < Suppose that C< is bounded on H(P). Then for every n € N,
lollop ^ \\cvhn\\ = -j-||(czm)ra|| = ffi^
pn pn
which gives
|c|nP
n
mn
1^1 /^mn_____ ^—r il
sup—-- < IIC^IIop.
neN pn
Conversely, suppose that (12) holds. Then for every function f (z) = ^TO=0 anzn € H(P), we have
ii^/ii2 =\\f(czm)\\2 = £Ki2M2ra/tra = EM2-0™ v *
n=O n=O P
VneN pn / n=o VneN pn
This shows that C^ is bounded on H(P) and also
iir* il <-IIC^Ilop ^ sup
n
mn
neN pn
From both necessity and sufficiency it follows that
|c|nP,
n
mn
1^1 /^mn n^v N
sup—--= yc^llop. >
neN Pn
The following is a consequence of Theorem 4.1. Corollary 4.2. Let c € D\ {0} and m € N \ {1}. If
lim sup ( ^mn ) ™ > —-
n—TO V Pn / |c|
then Cczm is unbounded on H(P). In particular, if
,. / Pmri i n
hmsup I —— I = oo,
n—TO \ Pn /
then Cczm is unbounded on H(P) for every 0 < |c| ^ 1.
< By the hypothesis, there exist a sequence (nk) f to and a number a > 0 such that
Anrifc i fc > a > _L for every fee N.
Pnfc y |c|
Hence,
|c|rafc/?mnfc > (a|c|)rafc ^ ^ ag k ^ Qo_
pnfc
From this and Theorem 4.1 the assertion follows. >
4.2. Some Notes on a General Symbol. In view of Corollary 4.2, we see that the situation for composition operators C^ on general weighted spaces H(P) is much more
complicated than on classical Hardy space H2(D) and Bergman space A2(D). More precisely, it is well-known that every holomorphic self-map of D with relatively compact image in D induces a bounded composition operator on these spaces. However, by Corollary 4.2, this is not valid for general weighted spaces H (5).
We end this section with the following sufficient conditions for boundedness of the operators C^ induced by such holomorphic self-maps of D.
Theorem 4.3. Let tp be a holomorphic self-map of B such that <£>(B) is compact in B. If
<x
32
E^n < (13)
n=0
then C^ is bounded on H (5).
< Take a number r € (0,1) so that <^(D) C rD. Then, for every f (z) = &nZra €
H(5), we have
00 / 00 r2n\ 2 / oo \ 2
n=0 \n=0 ßn / \n=0
where
i
^^ r2n \ 2 r2
Cr := I £ I < 00' since> ^ (1)> I™ sup—- ^ f2 < 1-
\ra=0 J fâ
Now writing Cf (z) = ^=0 bnzn and using the Cauchy integral formula, for every 5 € (0,1), we get
|bn| =
I(CV f )n(0)|
n!
^ 2vr
Cf (z)
T-n+1
dz
|z|=5
1
C
^-max|CV/(z)| sup 1/(^)1 ^ ^
which implies that |bn| ^ Cr ||f || for every n € N. Consequently,
OO \ 2
IC f II = £ |bn|2ßn < Cr If || £ A
OO \ 2
2
n I
Vra=0
Vra=0
From this the assertion follows. >
Remark 4.4. Condition (13) is essential but not sharp for Theorem 4.3. Indeed, for
' 1, n = 0, = { 1, n = 1 , 2, 3 mod 4,
n = 0 mod 4, n = 0,
(13) does not hold and, by Corollary 4.2, Cczm is unbounded on for every c € B \ {0}
and m € N \ {1}.
On the other hand, for 5«. = 1, (13) also does not hold, however in this case H(5) is Hardy space H2(D) and hence C^ is bounded on H(5) for every holomorphic function ^ with t£>(B) C B.
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Received June 8, 2020 Shanshan Hua
School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, E-mail: M170129@e.ntu.edu.sg;
Le Hai Khoi
School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, E-mail: lhkhoi@ntu.edu.sg https://orcid.org/0000-0002-4282-3449
Pham Trong Tien
University of Science, Vietnam National University,
334 Nguyen Trai St., Hanoi, Vietnam;
TIMAS, Thang Long University,
Nghiem Xuan Yem, Hoang Mai, Hanoi, Vietnam,
E-mail: phamt ien@vnu. edu. vn
https://orcid.org/0000-0001-5444-4393
Владикавказский математический журнал 2020, Том 22, Выпуск 3, С. 112-123
ОГРАНИЧЕННЫЕ КОМПОЗИЦИОННЫЕ ОПЕРАТОРЫ НА ВЕСОВЫХ ФУНКЦИОНАЛЬНЫХ ПРОСТРАНСТВАХ НА ЕДИНИЧНОМ КРУГЕ
Хуа С.1, Хой Л. Х.1, Тиен Ф. Ч.2>3
1 Наньянский технологический университет, Сингапур;
2 Вьетнамский национальный университет, Ханой, Вьетнам;
3 Тханг Лонг университет, Ханой, Вьетнам E-mail: M170129@e.ntu.edu.sg; lhkhoi@ntu.edu.sg; phamtien@vnu.edu.vn
Аннотация. Введен общий класс весовых пространств H (в) голоморфных функций в единичном круге D, содержащий в качестве частных случаев классические пространства Харди, Бергмана и Дири-хла. Полностью охарактеризована ограниченность на этих пространствах H (в) композиционных операторов C', порожденных аффинными и мономиальными символами р. Установлено также достаточное условие, при выполнении которого оператор Cv, порожденный символом р с относительно компактным образом p(D) в D, является ограниченным на H (в). Отменим, что в пространствах H (в) описание ограниченности композиционных операторов Cv зависит не только от функциональных свойств символов р, но и от поведения весовой последовательности в.
Ключевые слова: композиционный оператор, весовое пространство, весовая последовательность, голоморфные функции, единичный круг.
Mathematical Subject Classification (2010): 30D15, 47B37.
Образец цитирования: Hua, S., Khoi L. H. and Tien, P. T. Bounded Composition Operators on Weighted Function Spaces in the Unit Disk // Владикавк. мат. журн.—2020.—'Т. 22, № 3.—С. 112-123 (in English). DOI: 10.46698/p4238-0191-2122-t.
