New characterizations for differences of composition operators between weighted-type spaces in the unit ball
In this paper, we present some asymptotically equivalent expressions to the essential norm of differences of composition operators acting on weighted-type spaces of holomorphic functions in the unit ball of $\mathbb{C}^N$. Especially, the descriptions in terms of $\langle z, \zeta\rangle^m$ are desc...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507056389226496 |
|---|---|
| author | Chen, C. Chen, Cui Chen, C. |
| author_facet | Chen, C. Chen, Cui Chen, C. |
| author_sort | Chen, C. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2025-03-31T08:47:35Z |
| description | In this paper, we present some asymptotically equivalent expressions to the essential norm of differences of composition operators acting on weighted-type spaces of holomorphic functions in the unit ball of $\mathbb{C}^N$. Especially, the descriptions in terms of $\langle z, \zeta\rangle^m$ are described. From which the sufficient and necessary conditions of compactness follows immediately. Also, we characterize the boundedness of these operators. |
| doi_str_mv | 10.37863/umzh.v73i8.607 |
| first_indexed | 2026-03-24T02:03:15Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i8.607
UDC 517.5
C. Chen (Tianjin Univ. Finance and Economics, China)
NEW CHARACTERIZATIONS FOR DIFFERENCES OF COMPOSITION
OPERATORS BETWEEN WEIGHTED-TYPE SPACES IN THE UNIT BALL*
НОВI ХАРАКТЕРИСТИКИ РIЗНИЦЬ ОПЕРАТОРIВ КОМПОЗИЦIЇ
МIЖ ВАГОВИМИ ПРОСТОРАМИ В ОДИНИЧНIЙ КУЛI
We present some asymptotically equivalent expressions to the essential norm of differences of composition operators
acting on weighted-type spaces of holomorphic functions in the unit ball of \BbbC N . Especially, the descriptions in terms of
\langle z, \zeta \rangle m are described, from which the sufficient and necessary conditions of compactness follows immediately. Also, we
characterize the boundedness of these operators.
Запропоновано асимптотично еквiвалентнi вирази для суттєвої норми рiзниць операторiв композицiї, якi дiють у
вагових просторах голоморфних функцiй в одиничнiй кулi з \BbbC N . Зокрема, наведено опис у термiнах \langle z, \zeta \rangle m, з якого
безпосередньо випливають необхiднi та достатнi умови компактностi. Крiм того, охарактеризовано обмеженiсть цих
операторiв.
1. Introduction. Let \BbbC N denote the Euclidean space of complex dimension N(N \geq 1). For
z = (z1, . . . , zN ) and w = (w1, . . . , wN ) in \BbbC N , \langle z, w\rangle =
\sum N
j=1
zjwj and | z| =
\sqrt{}
\langle z, z\rangle . \BbbB is
the open unit ball of \BbbC N with boundary \partial \BbbB . H(\BbbB ) and S(\BbbB ) represent the class of holomorphic
functions and analytic self-maps on \BbbB , respectively. For \varphi ,\psi \in S(\BbbB ), the difference of composition
operator associated to \varphi and \psi is defined by (C\varphi - C\psi )f = f \circ \varphi - f \circ \psi for all f \in H(\BbbB ).
For 0 < \alpha <\infty , let H\infty
\alpha be the weighted-type space of holomorphic functions f on \BbbB satisfying
\| f\| \alpha = \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
(1 - | z| 2)\alpha | f(z)| <\infty .
With the norm \| f\| H\infty
\alpha
= | f(0)| + \| f\| \alpha , the weighted-type space becomes a Banach space.
For any point a \in \BbbB - \{ 0\} , the involutive automorphism \Phi a is defined by
\Phi a(z) =
a - Pa(z) - saQa(z)
1 - \langle z, a\rangle
, z \in \BbbB ,
where sa =
\sqrt{}
1 - | a| 2, and Pa(z) =
\langle z, a\rangle
| a| 2
a is the orthogonal projection from \BbbC N onto the one
dimensional subspace [a] generated by a, Qa(z) = z - Pa(z). When a = 0,\Phi a(z) = - z. It is
well-known that \Phi a interchanges the points 0 and a, that is, \Phi a(0) = a,\Phi a(a) = 0. For z, w \in \BbbB ,
the pseudohyperbolic distance between z and w is defined by \rho (z, w) = | \Phi w(z)| . For the simplicity,
we write \rho (z) = \rho (\varphi (z), \psi (z)).
Let X and Y be Banach spaces and T : X \rightarrow Y be a bounded linear operator. The essential norm
of T is the distance form T to the sets of compact operators, that is, \| T\| e,X\rightarrow Y = \mathrm{i}\mathrm{n}\mathrm{f}\{ \| T - K\| X\rightarrow Y :
K is compact from X to Y \} . Notice that \| T\| e = 0 if and only if the operator T is compact, so the
* This research was supported by the Research Project of Humanities and Social Sciences of Tianjin Municipal
Education Commission (Grant No. 2020SK102).
c\bigcirc C. CHEN, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8 1129
1130 C. CHEN
estimate on \| T\| e will lead to a condition for the operator T to be compact. For the results in this
topic, we refer the interested readers to the recent papers such as [1, 2, 9, 15, 17].
In 2009, Wulan et al. [18] (Theorem 2) obtained a new result about the compactness of compo-
sition operator on the classical Bloch space in the unit disk in terms of the sequence \{ zn\} \infty n=1. After
that, Ruhan Zhao [19] (Corollary 4.4) showed that \| C\varphi \| e,\scrB \alpha \rightarrow \scrB \beta \asymp \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}n\rightarrow \infty n\alpha - 1\| C\varphi zn\| \beta for
0 < \alpha , \beta < \infty . So, C\varphi : \scrB \alpha \rightarrow \scrB \beta is compact if and only if \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}n\rightarrow \infty n\alpha - 1\| C\varphi zn\| \beta = 0.
Subsequently to this, strong interest has arisen to describe some properties of composition operator
on Bloch-type spaces. For the results in the unit disk, one can refer to [4, 10, 13, 14, 18]. Then some
mathematicians have contributed to development of this new characterizations in the unit ball and
polydisk for some operators (see, e.g., [3, 5 – 8] and their references therein). In papers [11, 12, 16],
on the unit disk, such new descriptions for differences of classical linear operators was obtained. But
as far as we all known, there has no such characterizations for differences of any classical linear
operators in the unit ball, so these problems are in desired need of response. In this paper, we pay
our attention to start with the investigations for the differences of composition operators acting form
\alpha -weighted-type space to \beta -weighted-type space.
This paper is organized as follows. The boundedness of C\varphi - C\psi : H\infty
\alpha \rightarrow H\infty
\beta is exhibited in
Section 2 and then its essential norm is estimated in Section 3. In summary, this paper has systematic
exposition of equivalent conditions for the differences of composition operators from H\infty
\alpha to H\infty
\beta .
Throughout this paper, we will use the symbol C to denote a finite positive number, and it may
differ from one occurrence to the other. For two positive quantities A and B, the notations A \preceq B,
A \succeq B and A \asymp B mean that A \leq CB, A \geq CB and A/C \leq B \leq CA for some positive numbers
C, respectively. Besides, \BbbN denotes the set of all positive integers.
2. Boundedness of \bfitC \bfitvarphi - \bfitC \bfitpsi : \bfitH \infty
\bfitalpha \rightarrow \bfitH \infty
\bfitbeta . In this section, we give the characterization for
the boundedness of the operator C\varphi - C\psi : H\infty
\alpha \rightarrow H\infty
\beta . For any a \in \BbbB , we define the following
families test functions:
fa(z) =
(1 - | a| 2)\alpha
(1 - \langle z, a\rangle )2\alpha
and
g\varphi (a)(z) =
(1 - | \varphi (a)| 2)\alpha
(1 - \langle z, \varphi (a)\rangle )2\alpha
\langle \Phi \varphi (a)(z),\Phi \varphi (a)(\psi (a))\rangle
| \Phi \varphi (a)(\psi (a))|
,
g\psi (a)(z) =
(1 - | \psi (a)| 2)\alpha
(1 - \langle z, \psi (a)\rangle )2\alpha
\langle \Phi \psi (a)(z),\Phi \psi (a)(\varphi (a))\rangle
| \Phi \psi (a)(\varphi (a))|
.
It is easy to prove that \| g\varphi (a)\| H\infty
\alpha
\asymp \| g\psi (a)\| H\infty
\alpha
\preceq \| fa\| H\infty
\alpha
= 1. For the sake of convenience, we
use the notation as below
\scrT \beta
\alpha \varphi (z) =
(1 - | z| 2)\beta
(1 - | \varphi (z)| 2)\alpha
.
The main result in this section is the following theorem.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
NEW CHARACTERIZATIONS FOR DIFFERENCES OF COMPOSITION OPERATORS . . . 1131
Theorem 2.1. Let 0 < \alpha , \beta <\infty , \varphi , \psi \in S(\BbbB ). Then the following statements are equivalent:
(i) C\varphi - C\psi : H\infty
\alpha \rightarrow H\infty
\beta is bounded,
(ii1) \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
\scrT \beta
\alpha \varphi (z)\rho (z) + \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
\bigm| \bigm| \bigm| \scrT \beta
\alpha \varphi (z) - \scrT \beta
\alpha \psi (z)
\bigm| \bigm| \bigm| <\infty ,
(ii2) \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
\scrT \beta
\alpha \psi (z)\rho (z) + \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
\bigm| \bigm| \bigm| \scrT \beta
\alpha \varphi (z) - \scrT \beta
\alpha \psi (z)
\bigm| \bigm| \bigm| <\infty ,
(iii) \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\| (C\varphi - C\psi )fa\| H\infty
\beta
+ \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{
\| (C\varphi - C\psi )g\varphi (a)\| H\infty
\beta
, (C\varphi - C\psi )g\psi (a)\| H\infty
\beta
\Bigr\}
<\infty ,
(iv) \mathrm{s}\mathrm{u}\mathrm{p}
m\in \BbbN
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
<\infty .
In order to prove this result, we need some lemmas. For the first one, it was originally proved in
[19, 20].
Lemma 2.1. Let 0 < \alpha < \infty , m \in \BbbN and 0 \leq x \leq 1. Set rm =
\Bigl( m - 1
m - 1 + 2\alpha
\Bigr) 1/2
for m \geq 2
and rm = 0 for m = 1. Then Hm,\alpha (x) = xm - 1(1 - x2)\alpha has the following properties:
(i) \mathrm{m}\mathrm{a}\mathrm{x}
0\leq x\leq 1
Hm,\alpha (x) = Hm,\alpha (rm) =
\left\{
1, m = 1,\biggl(
m - 1
m - 1 + 2\alpha
\biggr) (m - 1)/2\biggl( 2\alpha
m - 1 + 2\alpha
\biggr) \alpha
, m \geq 2,
and \mathrm{l}\mathrm{i}\mathrm{m}m\rightarrow \infty m\alpha \mathrm{m}\mathrm{a}\mathrm{x}0\leq x\leq 1Hm,\alpha (x) =
\biggl(
2\alpha
e
\biggr) \alpha
,
(ii) for m \geq 1, Hm,\alpha is increasing on [0, rm] and decreasing on [rm, 1],
(iii) for m \geq 1, Hm,\alpha is decreasing on [rm, rm+1],
and \mathrm{m}\mathrm{i}\mathrm{n}
x\in [rm,rm+1]
Hm,\alpha (x) = Hm,\alpha (rm+1) =
\biggl(
m
m+ 2\alpha
\biggr) (m - 1)/2\biggl( 2\alpha
m+ 2\alpha
\biggr) \alpha
.
Consequently,
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
m\alpha \mathrm{m}\mathrm{i}\mathrm{n}
x\in [rm,rm+1]
Hm,\alpha (x) =
\biggl(
2\alpha
e
\biggr) \alpha
.
Lemma 2.2. Let 0 < \alpha <\infty ,m \in \BbbN . Then, for each \zeta \in \partial \BbbB , we have
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
m\alpha \| \langle \cdot , \zeta \rangle m\| H\infty
\alpha
=
\biggl(
2\alpha
e
\biggr) \alpha
. (2.1)
Proof. For any \zeta \in \partial \BbbB ,
\| \langle \cdot , \zeta \rangle m\| H\infty
\alpha
= \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
(1 - | z| 2)\alpha | \langle z, \zeta \rangle m| \leq \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
(1 - | z| 2)\alpha | z| m = \mathrm{s}\mathrm{u}\mathrm{p}
0\leq r\leq 1
(1 - r2)\alpha rm,
and, on the other hand,
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
(1 - | z| 2)\alpha | \langle z, \zeta \rangle | m \geq \mathrm{s}\mathrm{u}\mathrm{p}
0\leq r\leq 1
(1 - | r\zeta | 2)\alpha | \langle r\zeta , \zeta \rangle | m = \mathrm{s}\mathrm{u}\mathrm{p}
0\leq r\leq 1
(1 - r2)\alpha rm.
Thus,
m\alpha \| \langle \cdot , \zeta \rangle m\| H\infty
\alpha
= m\alpha \mathrm{s}\mathrm{u}\mathrm{p}
0\leq r\leq 1
(1 - r2)\alpha rm =
=
\biggl(
m
m+ 1
\biggr) \alpha
(m+ 1)\alpha \mathrm{s}\mathrm{u}\mathrm{p}
0\leq r\leq 1
(1 - r2)\alpha rm.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1132 C. CHEN
It follows from Lemma 2.1 (i) that (2.1) holds.
Lemma 2.2 is proved.
We will also make use of the following lemma. For the proof, see the original source [6].
Lemma 2.3. Let f \in H\infty
\alpha . Then
| (1 - | z| 2)\alpha f(z) - (1 - | w| 2)\alpha f(w)| \leq C\| f\| H\infty
\alpha
\rho (z, w)
for all z, w \in \BbbB .
Lemma 2.4. Let 0 < \alpha , \beta <\infty , \varphi , \psi \in S(\BbbB ). Then the following inequalities hold:
(i) \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
\scrT \beta
\alpha \varphi (z)\rho (z) \leq \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\| (C\varphi - C\psi )fa\| H\infty
\beta
+ \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\| (C\varphi - C\psi )g\varphi (a)\| H\infty
\beta
,
(ii) \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
\scrT \beta
\alpha \psi (z)\rho (z) \leq \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\| (C\varphi - C\psi )fa\| H\infty
\beta
+ \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\| (C\varphi - C\psi )g\psi (a)\| H\infty
\beta
,
(iii) \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
\bigm| \bigm| \bigm| \scrT \beta
\alpha \varphi (z) - \scrT \beta
\alpha \psi (z)
\bigm| \bigm| \bigm| \leq \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\| (C\varphi - C\psi )fa\| H\infty
\beta
+ \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\| (C\varphi - C\psi )g\psi (a)\| H\infty
\beta
.
Proof. For any a \in \BbbB , we have
\| (C\varphi - C\psi )f\varphi (a)\| H\infty
\beta
= \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
(1 - | z| 2)\beta | f\varphi (a)(\varphi (z)) - f\varphi (a)(\psi (z))| \geq
\geq (1 - | a| 2)\beta | f\varphi (a)(\varphi (a)) - f\varphi (a)(\psi (a))| \geq
\geq \scrT \beta
\alpha \varphi (a) -
(1 - | \varphi (a)| 2)\alpha (1 - | \psi (a)| 2)\alpha
| 1 - \langle \psi (a), \varphi (a)\rangle | 2\alpha
\scrT \beta
\alpha \psi (a)
and
\| (C\varphi - C\psi )g\varphi (a)\| H\infty
\beta
\geq (1 - | a| 2)\beta | g\varphi (a)(\varphi (a)) - g\varphi (a)(\psi (a))| =
= (1 - | a| 2)\beta (1 - | \varphi (a)| 2)\alpha
| 1 - \langle \psi (a), \varphi (a)\rangle | 2\alpha
\rho (a) =
=
(1 - | \varphi (a)| 2)\alpha (1 - | \psi (a)| 2)\alpha
| 1 - \langle \psi (a), \varphi (a)\rangle | 2\alpha
\scrT \beta
\alpha \psi (a)\rho (a).
Thus,
\scrT \beta
\alpha \varphi (a)\rho (a) \leq \| (C\varphi - C\psi )f\varphi (a)\| H\infty
\beta
\rho (a) + \| (C\varphi - C\psi )g\varphi (a)\| H\infty
\beta
\leq
\leq \| (C\varphi - C\psi )f\varphi (a)\| H\infty
\beta
+ \| (C\varphi - C\psi )g\varphi (a)\| H\infty
\beta
, (2.2)
where the last inequality follows from \rho (a) \leq 1. Analogously, we deduce that
\scrT \beta
\alpha \psi (a)\rho (a) \leq \| (C\varphi - C\psi )f\psi (a)\| H\infty
\beta
+ \| (C\varphi - C\psi )g\psi (a)\| H\infty
\beta
. (2.3)
Taking the supremum about a \in \BbbB in (2.2) and (2.3), we obtain
(i) \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\scrT \beta
\alpha \varphi (a)\rho (a) \leq \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\Bigl(
\| (C\varphi - C\psi )f\varphi (a)\| H\infty
\beta
+ \| (C\varphi - C\psi )g\varphi (a)\| H\infty
\beta
\Bigr)
\leq
\leq \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\| (C\varphi - C\psi )fa\| H\infty
\beta
+ \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\| (C\varphi - C\psi )g\varphi (a)\| H\infty
\beta
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
NEW CHARACTERIZATIONS FOR DIFFERENCES OF COMPOSITION OPERATORS . . . 1133
and
(ii) \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\scrT \beta
\alpha \psi (a)\rho (a) \leq \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\| (C\varphi - C\psi )fa\| H\infty
\beta
+ \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\| (C\varphi - C\psi )g\psi (a)\| H\infty
\beta
.
On the other hand, by Lemma 2.3 we note that
\| (C\varphi - C\psi )f\varphi (a)\| H\infty
\beta
\geq
\geq (1 - | a| 2)\beta | f\varphi (a)(\varphi (a)) - f\varphi (a)(\psi (a))| =
= (1 - | a| 2)\beta
\bigm| \bigm| \bigm| 1
(1 - | \varphi (a)| 2)\alpha
- (1 - | \varphi (a)| 2)\alpha
(1 - \langle \psi (a), \varphi (a)\rangle )2\alpha
\bigm| \bigm| \bigm| \geq
\geq
\bigm| \bigm| \bigm| \scrT \beta
\alpha \varphi (a) - \scrT \beta
\alpha \psi (a)
\bigm| \bigm| \bigm| - \bigm| \bigm| \bigm| \scrT \beta
\alpha \psi (a) -
(1 - | a| 2)\beta (1 - | \varphi (a)| 2)\alpha
(1 - \langle \psi (a), \varphi (a)\rangle )2\alpha
\bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \scrT \beta
\alpha \varphi (a) - \scrT \beta
\alpha \psi (a)
\bigm| \bigm| \bigm| -
- \scrT \beta
\alpha \psi (a)
\bigm| \bigm| \bigm| (1 - | \varphi (a)| 2)\alpha f\varphi (a)(\varphi (a)) - (1 - | \psi (a)| 2)\alpha f\varphi (a)(\psi (a))
\bigm| \bigm| \bigm| \succeq
\succeq
\bigm| \bigm| \bigm| \scrT \beta
\alpha \varphi (a) - \scrT \beta
\alpha \psi (a)
\bigm| \bigm| \bigm| - \scrT \beta
\alpha \psi (a)\rho (a). (2.4)
So together with (ii), we arrive at
(i) \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\bigm| \bigm| \bigm| \scrT \beta
\alpha \varphi (a) - \scrT \beta
\alpha \psi (a)
\bigm| \bigm| \bigm| \preceq \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\Bigl(
\| (C\varphi - C\psi )f\varphi (a)\| H\infty
\beta
+ \scrT \beta
\alpha \psi (a)\rho (a)
\Bigr)
\preceq
\preceq \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\| (C\varphi - C\psi )fa\| H\infty
\beta
+ \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\| (C\varphi - C\psi )g\psi (a)\| H\infty
\beta
.
Lemma 2.4 is proved.
Lemma 2.5. Let 0 < \alpha , \beta <\infty , \varphi , \psi \in S(\BbbB ). Then the following inequalities hold:
(i) \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\| (C\varphi - C\psi )fa\| H\infty
\beta
\preceq \mathrm{s}\mathrm{u}\mathrm{p}
m\in \BbbN
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
,
(ii) \mathrm{s}\mathrm{u}\mathrm{p}
a\in \BbbB
\mathrm{m}\mathrm{a}\mathrm{x}\{ \| (C\varphi - C\psi )g\varphi (a)\| H\infty
\beta
, (C\varphi - C\psi )g\psi (a)\| H\infty
\beta
\} \preceq
\preceq \mathrm{s}\mathrm{u}\mathrm{p}
m\in \BbbN
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
.
Proof. For \alpha > 0, recall that
1
(1 - \langle z, a\rangle )2\alpha
=
\infty \sum
k=0
\Gamma (k + 2\alpha )
k!\Gamma (2\alpha )
\langle z, a\rangle k,
then we express fa into Maclaurin expansion as follows:
fa(z) = (1 - | a| 2)\alpha
\infty \sum
k=0
\Gamma (k + 2\alpha )
k!\Gamma (2\alpha )
\langle z, a\rangle k.
If a = 0, fa(z) \equiv 1, (i) holds obvious. If a \not = 0, then
\| (C\varphi - C\psi )fa\| H\infty
\beta
\leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1134 C. CHEN
\leq (1 - | a| 2)\alpha
\infty \sum
k=0
\Gamma (k + 2\alpha )
k!\Gamma (2\alpha )
\| (C\varphi - C\psi )\langle \cdot , a\rangle k\| H\infty
\beta
\leq (2.5)
\leq (1 - | a| 2)\alpha
\infty \sum
k=0
\Gamma (k + 2\alpha )
k!\Gamma (2\alpha )
| a| kk - \alpha k\alpha \| (C\varphi - C\psi )\langle \cdot ,
a
| a|
\rangle k\| H\infty
\beta
\leq
\leq (1 - | a| 2)\alpha
\infty \sum
k=0
\Gamma (k + 2\alpha )
k!\Gamma (2\alpha )
| a| kk - \alpha \mathrm{s}\mathrm{u}\mathrm{p}
m\in \BbbN
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
. (2.6)
By Stirling’s formula,
\Gamma (k + \alpha )
k!\Gamma (\alpha )
\asymp k\alpha - 1 as k \rightarrow \infty . It follows that
\Gamma (k + 2\alpha )
k!\Gamma (2\alpha )
k - \alpha \asymp k\alpha - 1 as k \rightarrow \infty .
Hence,
\infty \sum
k=0
\Gamma (k + 2\alpha )
k!\Gamma (2\alpha )
| a| kk - \alpha \asymp
\infty \sum
k=0
k\alpha - 1| a| k \asymp
\infty \sum
k=0
\Gamma (k + \alpha )
k!\Gamma (\alpha )
| a| k \asymp 1
(1 - | a| )\alpha
, (2.7)
which combine with (2.6), we conclude (i).
Next, we prove the inequality (ii). When \varphi (a) = 0, g\varphi (a)(z) =
\langle z, \psi (a)\rangle
| \psi (a)|
, then
\| (C\varphi - C\psi )g\varphi (a)\| H\infty
\beta
=
\bigm\| \bigm\| \bigm\| \bigm\| (C\varphi - C\psi )
\biggl\langle
\cdot , \psi (a)
| \psi (a)|
\biggr\rangle \bigm\| \bigm\| \bigm\| \bigm\|
H\infty
\beta
\leq
\leq \mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
\| (C\varphi - C\psi )\langle \cdot , \zeta \rangle \| H\infty
\beta
\leq \mathrm{s}\mathrm{u}\mathrm{p}
m\in \BbbN
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
.
For \varphi (a) \not = 0,
g\varphi (a)(z) =
(1 - | \varphi (a)| 2)\alpha
(1 - \langle z, \varphi (a)\rangle )2\alpha
\langle \Phi \varphi (a)(z) - \Phi \varphi (a)(\psi (a)) + \Phi \varphi (a)(\psi (a)),\Phi \varphi (a)(\psi (a))\rangle
| \Phi \varphi (a)(\psi (a))|
=
= f\varphi (a)(z)\rho (a) + f\varphi (a)(z)
\langle \Phi \varphi (a)(z) - \Phi \varphi (a)(\psi (a)),\Phi \varphi (a)(\psi (a))\rangle
| \Phi \varphi (a)(\psi (a))|
,
thus, for any a \in \BbbB , we have
\| (C\varphi - C\psi )g\varphi (a)\| H\infty
\beta
\leq \| (C\varphi - C\psi )f\varphi (a)\| H\infty
\beta
+ 2\| (C\varphi - C\psi )f\varphi (a)\| H\infty
\beta
\preceq
\preceq \mathrm{s}\mathrm{u}\mathrm{p}
m\in \BbbN
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
. (2.8)
Here we used the fact that
\bigm| \bigm| \bigm| \bigm| \langle \Phi \varphi (a)(z) - \Phi \varphi (a)(\psi (a)),\Phi \varphi (a)(\psi (a))\rangle
| \Phi \varphi (a)(\psi (a))|
\bigm| \bigm| \bigm| \bigm| \leq 2.
Similarly, the inequality
\| (C\varphi - C\psi )g\psi (a)\| H\infty
\beta
\preceq \mathrm{s}\mathrm{u}\mathrm{p}
m\in \BbbN
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
(2.9)
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NEW CHARACTERIZATIONS FOR DIFFERENCES OF COMPOSITION OPERATORS . . . 1135
can easily be obtained by the methods used in the proof of (2.8). Taking the supremum about a \in \BbbB
in (2.8) and (2.9), (ii) comes ture.
Lemma 2.5 is proved.
Proof of Theorem 2.1. The implications (iv) \Rightarrow (iii) \Rightarrow (ii1) or (ii2) follow from Lemmas 2.4
and 2.5. We next prove (i)\Rightarrow (iv) and (ii)\Rightarrow (i).
(i) \Rightarrow (iv). Suppose that C\varphi - C\psi : H\infty
\alpha \rightarrow H\infty
\beta is bounded. For any m \in \BbbN and \zeta \in
\in \partial \BbbB , consider the function hm,\zeta (z) =
\langle z, \zeta \rangle m
\| \langle \cdot , \zeta \rangle m\| H\infty
\alpha
, then it is easy to see that hm,\zeta \in H\infty
\alpha with
\| hm,\zeta \| H\infty
\alpha
= 1. Note that from Lemma 2.2, there is a constant C > 0 independent of m and \zeta such
that \| \langle \cdot , \zeta \rangle m\| H\infty
\alpha
\leq Cm - \alpha . Combining with the boundedness of C\varphi - C\psi : H\infty
\alpha \rightarrow H\infty
\beta , it follows
that
\infty > \| C\varphi - C\psi \| H\infty
\alpha \rightarrow H\infty
\beta
\geq \| (C\varphi - C\psi )hm,\zeta \| H\infty
\beta
=
\| (C\varphi - C\psi )(\langle \cdot , \zeta \rangle m)\| H\infty
\beta
\| \langle \cdot , \zeta \rangle m\| H\infty
\alpha
\succeq
\succeq m\alpha \| (C\varphi - C\psi )(\langle \cdot , \zeta \rangle m)\| H\infty
\beta
,
for any m \in \BbbN and \zeta \in \partial \BbbB . Which shows the statement (i) \Rightarrow (iv).
(ii1) \Rightarrow (i). For any f \in H\infty
\alpha , we employ Lemma 2.3 to show that
\| (C\varphi - C\psi )f\| H\infty
\beta
= \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
(1 - | z| 2)\beta | f(\varphi (z)) - f(\psi (z))| \leq
\leq \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
(1 - | z| 2)\beta
(1 - | \varphi (z)| 2)\alpha
\bigm| \bigm| (1 - | \varphi (z)| 2)\alpha f(\varphi (z)) - (1 - | \psi (z)| 2)\alpha f(\psi (z))
\bigm| \bigm| +
+ \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
\bigm| \bigm| \bigm| \bigm| (1 - | z| 2)\beta (1 - | \psi (z)| 2)\alpha f(\psi (z))
(1 - | \varphi (z)| 2)\alpha
- (1 - | z| 2)\beta f(\psi (z))
\bigm| \bigm| \bigm| \bigm| \preceq
\preceq \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
\scrT \beta
\alpha \varphi (z)\rho (z) + \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
(1 - | \psi (z)| 2)\alpha | f(\psi (z))|
\bigm| \bigm| \bigm| \scrT \beta
\alpha \varphi (z) - \scrT \beta
\alpha \psi (z)
\bigm| \bigm| \bigm| \preceq
\preceq \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
\scrT \beta
\alpha \varphi (z)\rho (z) + \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbB
\bigm| \bigm| \bigm| \scrT \beta
\alpha \varphi (z) - \scrT \beta
\alpha \psi (z)
\bigm| \bigm| \bigm| <\infty . (2.10)
Thus, C\varphi - C\psi : H\infty
\alpha \rightarrow H\infty
\beta is bounded. Therefore, (i), (ii1), (iii), (iv) are equivalent. The
equivalence of statements (i), (ii2), (iii), (iv) can be proved in a similar manner.
Theorem 2.1 is proved.
3. Essential norm of \bfitC \bfitvarphi - \bfitC \bfitpsi : \bfitH \infty
\bfitalpha \rightarrow \bfitH \infty
\bfitbeta . In this section, we turn our attention to the
estimations for essential norm of C\varphi - C\psi : H\infty
\alpha \rightarrow H\infty
\beta . The proof of the main assertion relies on
the following two lemmas.
Lemma 3.1. Let 0 < \alpha , \beta <\infty , \varphi , \psi \in S(\BbbB ). Then the following inequalities hold:
(i) \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \varphi (z)| \rightarrow 1
\scrT \beta
\alpha \varphi (z)\rho (z) \preceq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| a| \rightarrow 1
\| (C\varphi - C\psi )fa\| H\infty
\beta
+ \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \varphi (a)| \rightarrow 1
\| (C\varphi - C\psi )g\varphi (a)\| H\infty
\beta
,
(ii) \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \psi (z)| \rightarrow 1
\scrT \beta
\alpha \psi (z)\rho (z) \preceq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| a| \rightarrow 1
\| (C\varphi - C\psi )fa\| H\infty
\beta
+ \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \psi (a)| \rightarrow 1
\| (C\varphi - C\psi )g\psi (a)\| H\infty
\beta
,
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1136 C. CHEN
(iii) \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
min\{ | \varphi (z)| ,| \psi (z)| \} \rightarrow 1
\bigm| \bigm| \bigm| \scrT \beta
\alpha \varphi (z) - \scrT \beta
\alpha \psi (z)
\bigm| \bigm| \bigm| \preceq
\preceq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| a| \rightarrow 1
\| (C\varphi - C\psi )fa\| H\infty
\beta
+ \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \psi (a)| \rightarrow 1
\| (C\varphi - C\psi )g\psi (a)\| H\infty
\beta
.
Proof. From the inequalities (2.2) – (2.4) the assertion follows easily.
Lemma 3.2. Let 0 < \alpha , \beta < \infty , \varphi , \psi \in S(\BbbB ), C\varphi - C\psi : H\infty
\alpha \rightarrow H\infty
\beta is bounded. Then the
following inequalities hold:
(i) \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| a| \rightarrow 1
\| (C\varphi - C\psi )fa\| H\infty
\beta
\preceq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
,
(ii) \mathrm{m}\mathrm{a}\mathrm{x}\{ \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \varphi (a)| \rightarrow 1
\| (C\varphi - C\psi )g\varphi (a)\| H\infty
\beta
, \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \psi (a)| \rightarrow 1
\| (C\varphi - C\psi )g\psi (a)\| H\infty
\beta
\} \preceq
\preceq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
.
Proof. For any a \in \BbbB and each positive integer N, employing (2.5) we obtain
\| (C\varphi - C\psi )fa\| H\infty
\beta
\leq (1 - | a| 2)\alpha
\infty \sum
k=0
\Gamma (k + 2\alpha )
k!\Gamma (2\alpha )
| a| k
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| (C\varphi - C\psi )
\biggl\langle
\cdot , a
| a|
\biggr\rangle k\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
H\infty
\beta
\leq
\leq (1 - | a| 2)\alpha
N\sum
k=0
\Gamma (k + 2\alpha )
k!\Gamma (2\alpha )
| a| k \mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
\| (C\varphi - C\psi )\langle \cdot , \zeta \rangle k\| H\infty
\beta
+
+ (1 - | a| 2)\alpha
\infty \sum
k=N+1
\Gamma (k + 2\alpha )
k!\Gamma (2\alpha )
| a| kk - \alpha \mathrm{s}\mathrm{u}\mathrm{p}
m\geq N+1
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
\triangleq
\triangleq J1 + J2.
For k \in \{ 0, 1, . . . , N\} , since \langle z, \zeta \rangle k \in H\infty
\alpha , for all \zeta \in \partial \BbbB and C\varphi - C\psi : H\infty
\alpha \rightarrow H\infty
\beta is bounded,
then
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
\| (C\varphi - C\psi )\langle \cdot , \zeta \rangle k\| H\infty
\beta
<\infty .
Hence,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| a| \rightarrow 1
J1 = 0.
On the other hand, noting (2.7) we have
J2 \preceq \mathrm{s}\mathrm{u}\mathrm{p}
m\geq N+1
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
,
which leads to
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| a| \rightarrow 1
J2 \preceq \mathrm{s}\mathrm{u}\mathrm{p}
m\geq N+1
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
.
Thus, (i) holds. Next based on the result in (2.8), it follows that
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NEW CHARACTERIZATIONS FOR DIFFERENCES OF COMPOSITION OPERATORS . . . 1137
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \varphi (a)| \rightarrow 1
\| (C\varphi - C\psi )g\varphi (a)\| H\infty
\beta
\preceq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \varphi (a)| \rightarrow 1
\| (C\varphi - C\psi )f\varphi (a)\| H\infty
\beta
\preceq
\preceq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| a| \rightarrow 1
\| (C\varphi - C\psi )fa\| H\infty
\beta
\preceq
\preceq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
.
Similarly, we can prove that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \varphi (a)| \rightarrow 1
\| (C\varphi - C\psi )g\psi (a)\| H\infty
\beta
\preceq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
.
Thus, we conclude (ii).
Lemma 3.2 is proved.
The following characterization about the essential norm of C\varphi - C\psi : H\infty
\alpha \rightarrow H\infty
\beta appears to be
useful for our purposes. For a proof, see Theorem 2 in [17].
Lemma 3.3. Let 0 < \alpha , \beta < \infty , \varphi , \psi \in S(\BbbB ) such that \mathrm{m}\mathrm{a}\mathrm{x}\{ \| \varphi 1\| \infty , \| \varphi 2\| \infty \} = 1.
If C\varphi , C\psi : H\infty
\alpha \rightarrow H\infty
\beta are bounded operators, then the essential norm \| C\varphi - C\psi \| e,H\infty
\alpha \rightarrow H\infty
\beta
is equivalent to the maximum of the following expressions:
(i) \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \varphi (z)| \rightarrow 1
\scrT \beta
\alpha \varphi (z)\rho (z),
(ii) \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \psi (z)| \rightarrow 1
\scrT \beta
\alpha \psi (z)\rho (z),
(iii) \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
min\{ | \varphi (z)| ,| \psi (z)| \} \rightarrow 1
\bigm| \bigm| \bigm| \scrT \beta
\alpha \varphi (z) - \scrT \beta
\alpha \psi (z)
\bigm| \bigm| \bigm| .
Theorem 3.1. Let 0 < \alpha , \beta < \infty , \varphi , \psi \in S(\BbbB ). If the operators C\varphi , C\psi : H\infty
\alpha \rightarrow H\infty
\beta are
bounded, then the following equivalences hold:
\| C\varphi - C\psi \| e,H\infty
\alpha \rightarrow H\infty
\beta
\approx
\approx \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \varphi (z)| \rightarrow 1
\scrT \beta
\alpha \varphi (z)\rho (z) + \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \psi (z)| \rightarrow 1
\scrT \beta
\alpha \psi (z)\rho (z) + \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
min\{ | \varphi (z)| ,| \psi (z)| \} \rightarrow 1
\bigm| \bigm| \bigm| \scrT \beta
\alpha \varphi (z) - \scrT \beta
\alpha \psi (z)
\bigm| \bigm| \bigm| \approx
\approx \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| a| \rightarrow 1
\| (C\varphi - C\psi )fa\| H\infty
\beta
+
+ \mathrm{m}\mathrm{a}\mathrm{x}\{ \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \varphi (a)| \rightarrow 1
\| (C\varphi - C\psi )g\varphi (a)\| H\infty
\beta
, \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \psi (a)| \rightarrow 1
\| (C\varphi - C\psi )g\psi (a)\| H\infty
\beta
\} \approx
\approx \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
.
Proof. The boundedness of C\varphi - C\psi : H\infty
\alpha \rightarrow H\infty
\beta comes easily from the boundedness of the
operators C\varphi and C\psi from H\infty
\alpha to H\infty
\beta . Thus, using the results in Lemmas 3.1 – 3.3, it suffices to
prove that
\| C\varphi - C\psi \| e,H\infty
\alpha \rightarrow H\infty
\beta
\succeq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
.
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1138 C. CHEN
Choose fm,\zeta (z) =
\langle z, \zeta \rangle m
\| \langle \cdot , \zeta \rangle m\| H\infty
\alpha
, then \| fm,\zeta \| H\infty
\alpha
= 1 and fm,\zeta \rightarrow 0,m\rightarrow \infty weakly in H\infty
\alpha . Thus,
for any compact operator K : H\infty
\alpha \rightarrow H\infty
\beta , we have \mathrm{l}\mathrm{i}\mathrm{m}m\rightarrow \infty \| fm,\zeta \| H\infty
\beta
= 0. Hence,
\| C\varphi - C\psi - K\| \geq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
\| (C\varphi - C\psi - K)fm,\zeta \| H\infty
\beta
\geq
\geq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
\| (C\varphi - C\psi )fm,\zeta \| H\infty
\beta
.
Then, from Lemma 2.2, we obtain
\| C\varphi - C\psi \| e,H\infty
\alpha \rightarrow H\infty
\beta
\geq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
\| (C\varphi - C\psi )fm,\zeta \| H\infty
\beta
\succeq
\succeq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
.
Theorem 3.1 is proved.
In view of Theorem 3.1, it gives equivalent conditions about the compactness of C\varphi - C\psi :
H\infty
\alpha \rightarrow H\infty
\beta .
Corollary 3.1. Let 0 < \alpha , \beta < \infty , \varphi , \psi \in S(\BbbB ). If the operators C\varphi , C\psi : H\infty
\alpha \rightarrow H\infty
\beta are
bounded, then the following conditions are equivalent:
(i) C\varphi - C\psi : H\infty
\alpha \rightarrow H\infty
\beta is compact,
(ii) \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \varphi (z)| \rightarrow 1
\scrT \beta
\alpha \varphi (z)\rho (z) + \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \psi (z)| \rightarrow 1
\scrT \beta
\alpha \psi (z)\rho (z) + \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
min\{ | \phi (z)| ,| \psi (z)| \} \rightarrow 1
\bigm| \bigm| \bigm| \scrT \beta
\alpha \varphi (z) - \scrT \beta
\alpha \psi (z)
\bigm| \bigm| \bigm| = 0,
(iii) \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| a| \rightarrow 1
\| (C\varphi - C\psi )fa\| H\infty
\beta
+
+\mathrm{m}\mathrm{a}\mathrm{x}\{ \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \varphi (a)| \rightarrow 1
\| (C\varphi - C\psi )g\varphi (a)\| H\infty
\beta
, \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| \psi (a)| \rightarrow 1
\| (C\varphi - C\psi )g\psi (a)\| H\infty
\beta
\} = 0,
(iv) \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in \partial \BbbB
m\alpha \| (C\varphi - C\psi )\langle \cdot , \zeta \rangle m\| H\infty
\beta
= 0.
References
1. J. Bonet, P. Domański, M. Lindström, Essential norm and weak compactness of composition operators on weithted
Banach spaces of analytic functions, Canad. Math. Bull., 42, № 2, 139 – 148 (1995).
2. J. Bonet, M. Lindström, E. Wolf, Difference of composition operators between weighted Banach spaces of holomorphic
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| id | umjimathkievua-article-607 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:03:15Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/bd/11dd9596308d8abf0dca2d1f460712bd.pdf |
| spelling | umjimathkievua-article-6072025-03-31T08:47:35Z New characterizations for differences of composition operators between weighted-type spaces in the unit ball New characterizations for differences of composition operators between weighted-type spaces in the unit ball New characterizations for differences of composition operators between weighted-type spaces in the unit ball Chen, C. Chen, Cui Chen, C. holomorphic functions In this paper, we present some asymptotically equivalent expressions to the essential norm of differences of composition operators acting on weighted-type spaces of holomorphic functions in the unit ball of $\mathbb{C}^N$. Especially, the descriptions in terms of $\langle z, \zeta\rangle^m$ are described. From which the sufficient and necessary conditions of compactness follows immediately. Also, we characterize the boundedness of these operators. Запропоновано асимптотично еквівалентні вирази для суттєвої норми різниць операторів композиції, які діють у вагових просторах голоморфних функцій в одиничній кулі з $\mathbb{C}^N.$&nbsp;&nbsp;Зокрема, наведено опис у термінах $\langle z, \zeta\rangle^m,$ з якого безпосередньо випливають необхідні та достатні умови компактності.&nbsp;&nbsp;Крім того, охарактеризовано обмеженість цих операторів. Institute of Mathematics, NAS of Ukraine 2021-08-18 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/607 10.37863/umzh.v73i8.607 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 8 (2021); 1129 - 1139 Український математичний журнал; Том 73 № 8 (2021); 1129 - 1139 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/607/9099 Copyright (c) 2021 Cui Chen |
| spellingShingle | Chen, C. Chen, Cui Chen, C. New characterizations for differences of composition operators between weighted-type spaces in the unit ball |
| title | New characterizations for differences of composition operators between weighted-type spaces in the unit ball |
| title_alt | New characterizations for differences of composition operators between weighted-type spaces in the unit ball New characterizations for differences of composition operators between weighted-type spaces in the unit ball |
| title_full | New characterizations for differences of composition operators between weighted-type spaces in the unit ball |
| title_fullStr | New characterizations for differences of composition operators between weighted-type spaces in the unit ball |
| title_full_unstemmed | New characterizations for differences of composition operators between weighted-type spaces in the unit ball |
| title_short | New characterizations for differences of composition operators between weighted-type spaces in the unit ball |
| title_sort | new characterizations for differences of composition operators between weighted-type spaces in the unit ball |
| topic_facet | holomorphic functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/607 |
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