Research Article | Open Access

# Composition Operators from the Weighted Bergman Space to the th Weighted Spaces on the Unit Disc

**Academic Editor:**Leonid Berezansky

#### Abstract

The boundedness of the composition operator from the weighted Bergman space to the recently introduced by the author, the th weighted space on the unit disc, is characterized. Moreover, the norm of the operator in terms of the inducing function and weights is estimated.

#### 1. Introduction

Let be the open unit disc in the complex plane , the Lebesgue area measure on , , , and the space of all analytic functions on the unit disc.

The weighted Bergman space , where and , consists of all such that With this norm, is a Banach space when , while for it is a FrΓ©chet space with the translation invariant metric

Let be a positive continuous function on a set (*weight*) and be fixed. The *th weighted-type space* on , denoted by consists of all such that

For the space becomes the weighted-type space , for the Bloch-type space , and for the Zygmund-type space .

For , the quantity is a seminorm on the th weighted-type space and a norm on , where is the set of all polynomials whose degrees are less than or equal to . A natural norm on the th weighted-type space can be introduced as follows: where is an element in . With this norm, the th weighted-type space becomes a Banach space.

For is obtained the space , on which a norm is introduced as follows: Some information on Zygmund-type spaces on the unit disc and some operators on them can be found, for example, in [1β6], for the case of the upper half-plane, see [7, 8], while some information in the setting of the unit ball can be found, for example, in [9β13]. This considerable interest in Zygmund-type spaces motivated us to introduce the th weighted-type space (see [8]).

Assume is a holomorphic self-map of . The composition operator induced by is defined on by

A typical problem is to provide function theoretic characterizations when induces bounded or compact composition operators between two given spaces of holomorphic functions. Some classical results on composition and weighted composition operators can be found, for example, in [14], while some recent results can be found in [1, 5, 7, 15β34] (see also related references therein).

Here we characterize the boundedness of the composition operator from the weighted Bergman space to the th weighted space on the unit disc when . The case was previously treated in [16, 22, 24, 31, 35]. Hence we will not consider this case here. See also [36] for some good results on weighted composition operators between weighted-type spaces. The case was treated, for example, in [26, 32]. For some other results on weighted composition operators which map a space into a weighted or a Bloch-type space, see, for example, [15, 17β21, 23, 25, 33, 34].

Let and be topological vector spaces whose topologies are given by translation-invariant metrics and , respectively, and be a linear operator. It is said that is *metrically bounded* if there exists a positive constant such that
for all . When and are Banach spaces, the metrically boundedness coincides with the usual definition of bounded operators between Banach spaces.

If is a Banach space, then the quantity is defined as follows: It is easy to see that this quantity is finite if and only if the operator is metrically bounded. For the case this is the standard definition of the norm of the operator , between two Banach spaces. If we say that an operator is bounded, it means that it is metrically bounded.

Throughout this paper, constants are denoted by , they are positive and may differ from one occurrence to the other. The notation means that there is a positive constant such that . Moreover, if both and hold, then one says that .

#### 2. Auxiliary Results

Here, we quote several auxiliary results. The first lemma is a direct consequence of a well-known estimate in [37, Propositionββ1.4.10]. Hence, we omit its proof.

Lemma 2.1. *Assume , , and . Then the function
**
belongs to . Moreover, .*

The next lemma is folklore and was essentially proved in [38]. We will sketch a proof of it for the completeness and the benefit of the reader.

Lemma 2.2. *Assume , , and . Then there is a positive constant independent of such that
*

*Proof. *By the subharmonicity of the function , , applied on the disk:
and since
we have that

From (2.5) and in light of the following well-known asymptotic relation [38]:
the lemma easily follows.

Lemma 2.3. *Assume and
**
Then .*

*Proof. *By using elementary transformations, we have
from which it follows that
which along with the fact implies the lemma.

We will also need the classical FaΓ di Bruno's formula where and the sum is over all nonnegative integers satisfying . For a nice exposition related to this formula see, for example, [39].

By using Bell polynomials , (2.10) can be written in the following form:

*Remark 2.4. *Since the summation in (2.11) is from 1 to . Moreover, since and , (2.11) can be written in the following form:

#### 3. Main Result

Here, we formulate and prove our main result.

Theorem 3.1. *Assume , , , is a weight on and is a holomorphic self-map of . Then is bounded if and only if
**
where for each fixed , the sum is over all nonnegative integers such that and .**Moreover, if the operator is bounded, then
*

*Remark 3.2. *Note that by (2.11) we see that the conditions in (3.1) can be written in the following form:

*Proof. *First assume that conditions in (3.1) hold. By formula (2.10) and Lemma 2.2 we have

From this, (2.2) with , and by conditions in (3.1), it follows that the operator is bounded. Moreover, if we consider the space , we have that

Now assume that the operator is bounded. For a fixed , and constants , set

Applying Lemma 2.1 we see that for every . Moreover, we have that

Now we show that for each , there are constants , such that
Indeed, by differentiating function , for each , the system in (3.8) becomes

By using Lemma 2.3 with , we obtain that the determinant of system (3.9) is different from zero from which the claim follows.

Now for each , we choose the corresponding family of functions which satisfy (3.8) and denote it by .

For each , the boundedness of the operator along with (2.10) and (3.7) implies that for each :
where (for each fixed ) the sum is over all nonnegative integers such that and .

From (3.10), it follows that for each ,

Now we use consecutively the test functions
in order to deal with the case . Note that

By applying (2.11) to the function we get
which along with the boundedness of the operator and (3.13) implies that
or equivalently (see Remark 2.4).

Further, by applying formula (2.11) to the function we get
From the boundedness of and (3.13), we get

From (3.16) and (3.17), and by using the triangle inequality it follows that

Using the fact and applying inequality (3.15) in (3.18) we get

Assume that we have proved the following inequalities:
for and a .

Applying formula (2.11) to the function , , we have that
From this, by using the boundedness of the operator , the boundedness of function , the triangle inequality, noticing that the coefficient at is independent of (it is equal ), and finally using hypothesis (3.20), we easily obtain
Hence, by induction, we get that (3.22) holds for each .

From (3.22) and bearing in mind Remark 2.4, for each fixed , we have that
where as usual for a fixed the sum is over all nonnegative integers such that and .

Hence from (3.11) and (3.23), we get

From (3.5) and (3.24), we obtain asymptotic relation (3.2).

#### Acknowledgment

The author would like to express his sincere thanks to the referees for numerous comments which improved the presentation of this paper.

#### References

- B. R. Choe, H. Koo, and W. Smith, βComposition operators on small spaces,β
*Integral Equations and Operator Theory*, vol. 56, no. 3, pp. 357β380, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Li and S. Stević, βVolterra-type operators on Zygmund spaces,β
*Journal of Inequalities and Applications*, vol. 2007, Article ID 32124, 10 pages, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S. Li and S. Stević, βGeneralized composition operators on Zygmund spaces and Bloch type spaces,β
*Journal of Mathematical Analysis and Applications*, vol. 338, no. 2, pp. 1282β1295, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Li and S. Stević, βProducts of Volterra type operator and composition operator from ${H}^{\infty}$ and Bloch spaces to Zygmund spaces,β
*Journal of Mathematical Analysis and Applications*, vol. 345, no. 1, pp. 40β52, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Li and S. Stević, βWeighted composition operators from Zygmund spaces into Bloch spaces,β
*Applied Mathematics and Computation*, vol. 206, no. 2, pp. 825β831, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. Zhu, βVolterra type operators from logarithmic Bloch spaces to Zygmund type spaces,β
*International Journal of Modern Mathematics*, vol. 3, no. 3, pp. 327β336, 2008. View at: Google Scholar | MathSciNet - S. Stević, βComposition operators from the Hardy space to the Zygmund-type space on the upper half-plane,β
*Abstract and Applied Analysis*, vol. 2009, Article ID 161528, 8 pages, 2009. View at: Google Scholar | MathSciNet - S. Stević, βComposition operators from the Hardy space to Zygmund-type spaces on the upper half-plane and the unit disk,β
*Journal of Computational Analysis and Applications*, vol. 12, 2010, (to appear). View at: Google Scholar - S. Li and S. Stević, βCesàro-type operators on some spaces of analytic functions on the unit ball,β
*Applied Mathematics and Computation*, vol. 208, no. 2, pp. 378β388, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Li and S. Stević, βIntegral-type operators from Bloch-type spaces to Zygmund-type spaces,β
*Applied Mathematics and Computation*, vol. 215, no. 2, pp. 464β473, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Stević, βOn an integral operator from the Zygmund space to the Bloch-type space on the unit ball,β
*Glasgow Mathematical Journal*, vol. 51, no. 2, pp. 275β287, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. Zhu,
*Spaces of Holomorphic Functions in the Unit Ball*, vol. 226 of*Graduate Texts in Mathematics*, Springer, New York, NY, USA, 2005. View at: MathSciNet - X. Zhu, βExtended Cesàro operators from ${H}^{\infty}$ to Zygmund type spaces in the unit ball,β
*Journal of Computational Analysis and Applications*, vol. 11, no. 2, pp. 356β363, 2009. View at: Google Scholar | MathSciNet - C. C. Cowen and B. D. MacCluer,
*Composition Operators on Spaces of Analytic Functions*, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1995. View at: MathSciNet - X. Fu and X. Zhu, βWeighted composition operators on some weighted spaces in the unit ball,β
*Abstract and Applied Analysis*, vol. 2008, Article ID 605807, 8 pages, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet - D. Gu, βWeighted composition operators from generalized weighted Bergman spaces to weighted-type spaces,β
*Journal of Inequalities and Applications*, vol. 2008, Article ID 619525, 14 pages, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S. Li and S. Stević, βWeighted composition operators from $\alpha $-Bloch space to ${H}^{\infty}$ on the polydisc,β
*Numerical Functional Analysis and Optimization*, vol. 28, no. 7-8, pp. 911β925, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Li and S. Stević, βWeighted composition operators from ${H}^{\infty}$ to the Bloch space on the polydisc,β
*Abstract and Applied Analysis*, vol. 2007, Article ID 48478, 13 pages, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Li and S. Stević, βWeighted composition operators between ${H}^{\infty}$ and $\alpha $-Bloch spaces in the unit ball,β
*Taiwanese Journal of Mathematics*, vol. 12, no. 7, pp. 1625β1639, 2008. View at: Google Scholar | MathSciNet - S. Ohno, K. Stroethoff, and R. Zhao, βWeighted composition operators between Bloch-type spaces,β
*The Rocky Mountain Journal of Mathematics*, vol. 33, no. 1, pp. 191β215, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Stević, βComposition operators between ${H}^{\infty}$ and $a$-Bloch spaces on the polydisc,β
*Zeitschrift für Analysis und ihre Anwendungen*, vol. 25, no. 4, pp. 457β466, 2006. View at: Google Scholar | MathSciNet - S. Stević, βWeighted composition operators between mixed norm spaces and ${H}_{\alpha}^{\infty}$ spaces in the unit ball,β
*Journal of Inequalities and Applications*, vol. 2007, Article ID 28629,, 9 pages, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S. Stević, βEssential norms of weighted composition operators from the $\alpha $-Bloch space to a weighted-type space on the unit ball,β
*Abstract and Applied Analysis*, vol. 2008, Article ID 279691, 11 pages, 2008. View at: Google Scholar | MathSciNet - S. Stević, βNorms of some operators from Bergman spaces to weighted and Bloch-type spaces,β
*Utilitas Mathematica*, vol. 76, pp. 59β64, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S. Stević, βNorm of weighted composition operators from Bloch space to ${H}_{\mu}^{\infty}$ on the unit ball,β
*Ars Combinatoria*, vol. 88, pp. 125β127, 2008. View at: Google Scholar | MathSciNet - S. Stević, βWeighted composition operators from mixed norm spaces into weighted Bloch spaces,β
*Journal of Computational Analysis and Applications*, vol. 11, no. 1, pp. 70β80, 2009. View at: Google Scholar | MathSciNet - S.-I. Ueki, βComposition operators on the Privalov spaces of the unit ball of ${\u2102}^{n}$,β
*Journal of the Korean Mathematical Society*, vol. 42, no. 1, pp. 111β127, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S.-I. Ueki, βWeighted composition operators on the Bargmann-Fock space,β
*International Journal of Modern Mathematics*, vol. 3, no. 3, pp. 231β243, 2008. View at: Google Scholar | MathSciNet - S. I. Ueki, βWeighted composition operators on some function spaces of entire functins,β to appear in
*Bulletin of the Belgian Mathematical Society. Simon Stevin*. View at: Google Scholar - S.-I. Ueki and L. Luo, βCompact weighted composition operators and multiplication operators between Hardy spaces,β
*Abstract and Applied Analysis*, vol. 2008, Article ID 196498, 12 pages, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. Wolf, βWeighted composition operators between weighted Bergman spaces and weighted Banach spaces of holomorphic functions,β
*Revista Matemática Complutense*, vol. 21, no. 2, pp. 475β480, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet - E. Wolf, βWeighted composition operators between weighted Bergman spaces and weighted Bloch type spaces,β
*Journal of Computational Analysis and Applications*, vol. 11, no. 2, pp. 317β321, 2009. View at: Google Scholar | MathSciNet - W. Yang, βWeighted composition operators from Bloch-type spaces to weighted-type spaces,β to appear in
*Ars Combinatoria*. View at: Google Scholar - X. Zhu, βWeighted composition operators from $F(p,q,s)$ spaces to ${H}_{\mu}^{\infty}$ spaces,β
*Abstract and Applied Analysis*, vol. 2009, Article ID 290978, 14 pages, 2009. View at: Google Scholar | MathSciNet - X. Zhu, βWeighted composition operators between ${H}^{\infty}$ and Bergman type spaces,β
*Communications of the Korean Mathematical Society*, vol. 21, no. 4, pp. 719β727, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet - A. Montes-Rodríguez, βWeighted composition operators on weighted Banach spaces of analytic functions,β
*Journal of the London Mathematical Society*, vol. 61, no. 3, pp. 872β884, 2000. View at: Google Scholar | Zentralblatt MATH | MathSciNet - W. Rudin,
*Function Theory in the Unit Ball of*, vol. 241 of*ℂ*^{n}*Fundamental Principles of Mathematical Science*, Springer, New York, NY, USA, 1980. View at: MathSciNet - T. M. Flett, βThe dual of an inequality of Hardy and Littlewood and some related inequalities,β
*Journal of Mathematical Analysis and Applications*, vol. 38, pp. 746β765, 1972. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Wikipedia, http://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%E2%80%99s_formula.

#### Copyright

Copyright © 2009 Stevo Stević. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.