Inequalities Involving Essential Norm Estimates of Product-Type Operators

Abstract

Consider an open unit disk in the complex plane , a holomorphic function on , and a holomorphic self-map of . For an analytic function , the weighted composition operator is denoted and defined as follows: . We estimate the essential norm of this operator from Dirichlet-type spaces to Bers-type spaces and Bloch-type spaces.

1. Introduction and Preliminaries

Consider an open unit disk in the complex plane . Let denote the class of all analytic functions on , be the class of all holomorphic self-maps of , and be the space of all bounded holomorphic functions on . Let and be a holomorphic self-map of . For , the composition operator and multiplication operator are, respectively, defined by

The weighted composition operator is denoted and defined aswhere is a product-type operator as . Clearly, this operator can be seen as a generalization of the composition operator and multiplication operator. It can be easily seen that, for , the operator reduced to . If , the operator gets reduced to . This operator is basically a linear transformation of defined by , for in and in . The basic aim is to give the operator-theoretic characterization of these operators in terms of function-theoretic characterization of their including functions. Various holomorphic function spaces on various domains have been studied for the boundedness and compactness of weighted composition operators acting on them. Moreover, a number of papers have been studied on these operators acting on different spaces of holomorphic functions on various domains. For more details, see [1–14] and the references therein. We say that a linear operator is bounded if the image of a bounded set is a bounded set. Moreover, a linear operator is said to be compact if it maps the bounded sets to those sets whose closure is compact. For each , the weighted Bloch space is defined as follows:

In this expression, seminormed is defined. This space forms a Banach space with the natural norm defined by

For , this space gets reduced to classical Bloch space. A function is said to be a weight if it is continuous. For , the weight is said to be radial if . A weight is said to be a standard weight if . For a weight function , the Bloch-type space is defined by

The little Bloch-type space is the closure of the set of polynomials in and is defined as follows:

Both and form a Banach space with the following norm:

For more information about these spaces, one may refer [1–3, 5, 6, 15, 16] and the references therein. Likewise, for weight , the Bers-type space is defined as follows:

It is a nonseparable Banach space with the norm . The closure of the set of polynomials in forms a separable Banach space. This set is denoted by and is defined as

These spaces and their properties are discussed in many papers; some of these are [3, 15, 16] and the references therein. The Dirichlet space is defined as follows:where denotes the normalized Lebesgue area measure on . With the following norm, it is a Hilbert space:

Consider a function which is right continuous and increasing. In this paper, we consider function as a weight function. With a weight function , the Dirichlet type space is given as follows:

Clearly, space forms a Hilbert space with the norm defined by

Here, we have , , and gives , that is, the usual Dirichlet-type space. This gives a classical Dirichlet space for a case when and for , and we gain the Hardy space . These spaces have been studied widely in various papers. For example, Aleman, in [17], obtained that each element of can be written as a quotient of two bounded functions in . Kerman and Sawyer [18], by taking some conditions on weight function , characterized Carleson measures and multipliers of in terms of maximal operator.

The Möbius invariant space generated by is denoted by . The space contains those functions which satisfy the following:where is the Möbius transformation of . Wulan and Zhu, in [19], characterized Lacunary series in the space under some conditions on weight function . Furthermore, Wulan and Zhou [20] characterized space in terms of fractional-order derivatives of function. They also established a relationship between Morrey type spaces and space in terms of fractional order derivatives. In the study of spaces, the following two conditions play a very important role:where

Let be the class of multipliers of , that is,

Bao et al., in [21], characterized the interpolating sequences for of space , under certain conditions of weight function . They also obtained corona theorem, - equation, and corona-type decomposition theorem on . For more details, see [9, 15, 21–25] and the references therein.

From [26], one can see that if satisfies (1), then

If satisfies (16), then

From condition (16), we get that for . Also, there exist sufficiently small for which is increasing and is decreasing. For more information about weight function , one can refer [19–21].

The criterion of boundedness as well as compactness has been discussed in many papers. Recently, Gürbüz, in [27], studied the boundedness of generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively, and, in [28], he investigated the generalized weighted Morrey estimates for the boundedness of Marcinkiewicz integrals with rough kernel associated with Schrödinger operator and their commutators. Furthermore, in [29], Gürbuüz studied the behavior of multi-sublinear fractional maximal operators and rough multilinear fractional integral both on product and weighted spaces and, in [30], he obtained the boundedness of the variation and oscillation operators for the family of multilinear integrals with Lipschitz functions on weighted Morrey spaces. Among others in [3], we obtained the following results about boundedness and compactness of given as follows.

Theorem 1. Let and be two weight functions, , and be a self-holomorphic map on . Then, the operator is bounded if and only if the following conditions hold:(i)(ii)

Furthermore, if the operator is bounded, then

Theorem 2. Let be a standard weight, , and be a self-holomorphic map on . Let be a weight function. Assume that is bounded. Then, the operator is compact if and only if the following conditions hold:(i)(ii)

Theorem 3. Let be a weight and be a weight function, , and be a self-holomorphic map on . Then, the operator is bounded if and only if the following condition holds:

Theorem 4. Let be a standard weight, , and be a self-analytic map on . Let be a weight function. Assume that the operator is bounded. Then, is compact if and only if the following condition is satisfied:

The aim of this paper is to provide some estimates of essential norm of the operator as well as of .

Assume that is a bounded linear operator for Banach spaces and . The essential norm of operator is denoted and defined as follows:where is the operator norm. In other words, the essential norm is the distance from compact operators mapping into to the bounded linear operator . If , that is, the two Banach spaces are same, then the norm is simply denoted by . For unbounded linear operator , we have . As the class of all compact operators is contained in the class of all bounded operators, in fact, this subset is closed, which implies that the operator is compact if and only if . Thus, the estimate of essential norm leads to the compactness of the operator. Various results on the essential norm of different operators such as multiplication, composition, differentiation, weighted composition, generalized weighted composition, and their different combinations are studied in numerous research papers, and some of the references are [31–37].

This study is formulated in a systematic way. Introduction and literature part is kept in Section 1. In Section 2, we estimated the essential norm of operator . Finally, in Section 3, we estimated the essential norm of operator . Throughout the paper, the notation , for any two positive quantities and , which means that , where is some positive constant. The value of constant may change from one place to the other. We write if and .

2. Essential Norm of Weighted Composition Operator from Dirichlet-Type Space to Bloch-Type Space

Theorem 5. Let be a standard weight, , and be a self-analytic map on . Let be a weight function. Assume that is bounded. Then,where

Proof. At first, we show thatFor , define a functionwhere and . It can be easily checked that and, for all , . On calculation, we have and . Furthermore, on compact subsets of , converges to zero as . Hence, for any compact operator and any such that , we obtainIn the above inequality, take on both sides, and we obtainAgain, for , define another function:where and . In the similar manner, we can check that and, for all , . On calculation, we have and . Furthermore, on compact subsets of , converges to zero as . Thus, for any compact operator and any such that , we obtainBy taking on both sides of the above inequality, we obtainOn applying the definition of essential norm, we find thatNext, we prove thatFor , consider , defined as follows:Clearly, is compact on and . Consider a sequence satisfying as . Then, for all , operator is compact. By using the definition of essential norm, we obtainTherefore, we only have to prove thatLet be a function in satisfying ; then, we haveClearly, . Furthermore, consider a large enough such that, for all , we have . Thus, we obtainwhereTaking the operator to 1 and and using its boundedness, it easily follows that andAlso, on compact subsets of , uniformly converges to as ; thus, we haveSimilarly, for and the fact that converges uniformly to on compact subsets of as , we obtainNow, consider . We have , whereFirst, we consider . As , we obtainOn taking limit as , we obtainIn the similar manner, we obtainOn combining the above two inequalities, we obtainNext, consider . We have , whereBy similar calculation, we obtainOn taking limit , we obtainIn the similar manner, we obtainCombining the above two inequalities, we obtainOn combining (40), (43), (44), (49), and (54), we obtainThus, inequalities (37) and (55) imply thatHence, inequalities (34) and (56) complete the theorem.
The following corollary can be easily obtained from Theorem 5.

Corollary 1. Let be a standard weight and be a self-analytic map on . Let be a weight function. Assume that is bounded. Then,

3. Essential Norm of Weighted Composition Operator from Dirichlet-Type Space to Bers-Type Space

In this section, we consider the Bers-type spaces and estimated the essential norm of weighted composition operator from to .

Theorem 6. Let be a standard weight, , and be a self-analytic map on . Let be a weight function. Assume that the operator is bounded. Then,

Proof. Firstly, we prove thatConsider a function such that , and on compact subsets of , converges to zero as . Thus, for any compact operator and any such that , we obtainTaking on both sides, we obtainOn applying the definition of essential norm, we find thatFinally, we prove thatFor this, consider with and a sequence satisfying as defined in Theorem 5. Then, for all , the operator is compact. By using the definition of essential norm, we obtainSo, we only have to prove thatLet be a function in satisfying ; then, we haveFurthermore, consider a large enough such that, for all , we have . Thus, we obtainwhereSimilar to Theorem 5, for and the fact that converges uniformly to on compact subsets of as , we obtainNext, we consider . We have , whereOn calculation, we obtainTaking , we obtainSimilarly,On combining the above two inequalities, we obtainFrom inequalities (67), (69), and (74), we obtainThus, inequalities (64) and (75) imply thatHence, inequalities (62) and (76) complete the theorem.
The following corollary can be easily obtained from Theorem 6.

Corollary 2. Let be a standard weight and be a self-analytic map on . Let be a weight function. Assume that the operator is bounded. Then,

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.