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Generalized Stević-Sharma operators from the minimal Möbius invariant space into Bloch-type spaces

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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

The aim of this study is to investigate the boundedness, essential norm, and compactness of generalized Stević-Sharma operator from the minimal Möbius invariant space into Bloch-type space.

Keywords: generalized Stević-Sharma operator; minimal Möbius invariant space; Bloch-type space; essential norm
MSC 2010: 47B38; 30H25; 30H30

1 Introduction

Let D be the open unit disk in the complex plane C and N the set of positive integers. Denote by H ( D ) the class of all analytic functions on D and S ( D ) the family of all analytic self-maps of D .

The set of all conformal automorphisms of D forms a group, called the Möbius group, and is denoted by Aut( D ). It is well known from complex analysis that every element of Aut( D ) has the form e i θ σ w ( z ) , where θ is a real number and

σ w ( z ) = w − z 1 − w ¯ z , w ∈ D ,

is a special automorphism of D exchanging the points w and 0. Let X be a linear space of analytic functions on D . Then, X is said to be Möbius invariant if for all f ∈ X and ν ∈ Aut( D ), f ∘ ν ∈ X and satisfies that ‖ f ∘ ν ‖ X = ‖ f ‖ X (see [1]). A typical example of Möbius invariant space is the analytic Besov space B p . Recall that for 1 < p < ∞ , a function f ∈ H ( D ) belongs to B p if

∫ D ∣ f ′ ( z ) ∣ p ( 1 − ∣ z ∣ 2 ) p − 2 d A ( z ) < ∞ ,

where d A is the normalized Lebesgue area measure on D . Note that when p = 2 , B 2 is known as the Dirichlet space, which is the only Möbius invariant Hilbert space (see [2]).

The analytic Besov space B 1 consists of all f ∈ H ( D ) , which have a representation as:

f ( z ) = ∑ n = 1 ∞ a n σ λ n ( z ) ,

for some sequences { a n } n ∈ N ∈ l 1 and { λ n } n ∈ N in D . The norm in B 1 is defined by:

‖ f ‖ B 1 = inf ∑ n = 1 ∞ ∣ a n ∣ : f ( z ) = ∑ n = 1 ∞ a n σ λ n ( z ) .

By [1], we know that the space B 1 is the minimal Möbius invariant space, as it is contained in any Möbius invariant space. Furthermore, B 1 is identical with the set of f ∈ H ( D ) for which f ″ ∈ L 1 ( D , d A ) , and there exist constants C 1 and C 2 such that

C 1 ‖ f ‖ B 1 ≤ ∣ f ( 0 ) ∣ + ∣ f ′ ( 0 ) ∣ + ∫ D ∣ f ″ ( z ) ∣ d A ( z ) ≤ C 2 ‖ f ‖ B 1 .

For more studies of B 1 space, see also [3–8].

Suppose that μ is a weight, namely, a strictly positive continuous function on D . We also assume that μ is radial: μ ( z ) = μ ( ∣ z ∣ ) for any z ∈ D . An f ∈ H ( D ) is said to belong to the Bloch-type space ℬ μ , if

sup z ∈ D μ ( z ) ∣ f ′ ( z ) ∣ < ∞ .

ℬ μ is a Banach space under the norm ‖ f ‖ ℬ μ = ∣ f ( 0 ) ∣ + sup z ∈ D μ ( z ) ∣ f ′ ( z ) ∣ . When μ ( z ) = 1 − ∣ z ∣ 2 , the induced space ℬ μ reduces to the classical Bloch space, which is the maximal Möbius invariant space [9]. For some results on the Bloch-type spaces and operators on them, see, for instance, [4,10–14].

Suppose that φ ∈ S ( D ) and u ∈ H ( D ) , the composition and multiplication operators on H ( D ) are defined, respectively, by:

C φ f ( z ) = f ( φ ( z ) ) and M u f ( z ) = u ( z ) f ( z ) ,

where f ∈ H ( D ) and z ∈ D . The product of these two operators is known as the weighted composition operator W u , φ = u ( z ) f ( φ ( z ) ) . It is important to provide function theoretic characterizations when φ and u induce a bounded or compact weighted composition operator on various function spaces. See [7,15] for more research about the (weighted) composition operators acting on several spaces of analytic functions. The differentiation operator D , which is defined by D f ( z ) = f ′ ( z ) for f ∈ H ( D ) , plays an important role in operator theory and dynamical system.

The first papers on product-type operators including the differentiation operator dealt with the operators D C φ and C φ D (see, for example, [11,16–19]). In [20,21], Stević and co-workers introduced the so-called Stević-Sharma operator as follows:

T u , v , φ f ( z ) = u ( z ) f ( φ ( z ) ) + v ( z ) f ′ ( φ ( z ) ) , f ∈ H ( D ) ,

where u , v ∈ H ( D ) and φ ∈ S ( D ) . By taking some specific choices of the involving symbols, we can easily obtain the general product-type operators:

M u C φ = T u , 0 , φ , C φ M u = T u ∘ φ , 0 , φ , M u D = T 0 , u , i d , D M u = T u ′ , u , i d , C φ D = T 0 , 1 , φ , D C φ = T 0 , φ ′ , φ , M u C φ D = T 0 , u , φ , M u D C φ = T 0 , u φ ′ , φ , C φ M u D = T 0 , u ∘ φ , φ , D M u C φ = T u ′ , u φ ′ , φ , C φ D M u = T u ′ ∘ φ , u ∘ φ , φ , D C φ M u = T φ ′ ( u ′ ∘ φ ) , φ ′ ( u ∘ φ ) , φ .

Recently, there has been an increasing interest in studying the Stević-Sharma operator between various spaces of analytic function. For instance, the boundedness, compactness, and essential norm of T u , v , φ on the weighted Bergman space were characterized by Stević et al. in [20,21]. Wang et al. in [22] considered the difference of two Stević-Sharma operators and investigated its boundedness, compactness, and order boundedness between Banach spaces of analytic functions. Zhu et al. in [14] provided some necessary and sufficient conditions for T u , v , φ to be bounded or compact when considered as an operator from the analytic Besov space B p into Bloch space. Abbasi et al. in [23] generalized the Stević-Sharma operator as follows:

T u , v , φ m f ( z ) = u ( z ) f ( φ ( z ) ) + v ( z ) f ( m ) ( φ ( z ) ) , m ∈ N ,

and studied its boundedness, compactness, and essential norm from Hardy space into the n th weighted-type space, which was introduced by Stević in [24] (see also [25]). Note that when m = 1 , we obtain the Stević-Sharma operator T u , v , φ . Some more related results can be found (see, e.g., [4, 5,8,10–14,26–32] and references therein).

Motivated by the aforementioned studies, here we investigate the boundedness and essential norm of the generalized Stević-Sharma operator T u , v , φ m from the minimal Möbius invariant space B 1 into the Bloch-type space ℬ μ . As a corollary, we give the characterizations of its compactness.

Recall that the essential norm of a bounded linear operator T : X → Y is the distance from T to the compact operators K : X → Y , that is,

‖ T ‖ e , X → Y = inf { ‖ T − K ‖ X → Y : K is compact } ,

where X and Y are the Banach spaces. Note that ‖ T ‖ e , X → Y = 0 if and only if T : X → Y is compact.

Throughout this article, for nonnegative quantities X and Y , we use the abbreviation X ≲ Y or Y ≳ X if there exists a positive constant C independent of X and Y such that X ≤ C Y . Moreover, we write X ≈ Y if X ≲ Y ≲ X .

2 Auxiliary results

In this section, we state several auxiliary results that are needed in the proofs of our main results. The following lemma can be found, for example, in [8] (see also [33]).

Lemma 1

Let k ∈ N , then

‖ f ‖ ∞ ≲ ‖ f ‖ B 1 a n d ( 1 − ∣ z ∣ 2 ) k ∣ f ( k ) ( z ) ∣ ≲ ‖ f ‖ B 1

for each f ∈ B 1 .

For any w ∈ D and j ∈ N , set

(1) f j , w ( z ) = ( 1 − ∣ w ∣ 2 ) j ( 1 − w ¯ z ) j , z ∈ D .

It is easily seen that f j , w ∈ B 1 and sup w ∈ D ‖ f j , w ‖ B 1 ≲ 1 for each j ∈ N . Moreover, f j , w converges to 0 uniformly on compact subsets of D as ∣ w ∣ → 1 .

Lemma 2

Let m ∈ N and m > 1 . For any w ∈ D \ { 0 } and i , k ∈ { 0 , 1 , m , m + 1 } , there exists a function g i , w ∈ B 1 such that

g i , w ( k ) ( w ) = w ¯ k δ i k ( 1 − ∣ w ∣ 2 ) k ,

where δ i k is the Kronecker delta.

Proof

For any w ∈ D \ { 0 } and constants c 1 , c 2 , c 3 , and c 4 , let

g w ( z ) = ∑ j = 1 4 c j f j , w ( z ) ,

where f j , w is defined in (1). For each i ∈ { 0 , 1 , m , m + 1 } , the system of linear equations

g w ( w ) = c 1 + c 2 + c 3 + c 4 = δ i 0 , g w ′ ( w ) = ( c 1 + 2 c 2 + 3 c 3 + 4 c 4 ) w ¯ 1 − ∣ w ∣ 2 = w ¯ δ i 1 1 − ∣ w ∣ 2 , g w ( m ) ( w ) = m ! c 1 + ( m + 1 ) ! c 2 + ( m + 2 ) ! 2 c 3 + ( m + 3 ) ! 6 c 4 w ¯ m ( 1 − ∣ w ∣ 2 ) m = w ¯ m δ i m ( 1 − ∣ w ∣ 2 ) m , g w ( m + 1 ) ( w ) = ( m + 1 ) ! c 1 + ( m + 2 ) ! c 2 + ( m + 3 ) ! 2 c 3 + ( m + 4 ) ! 6 c 4 w ¯ m + 1 ( 1 − ∣ w ∣ 2 ) m + 1 = w ¯ m + 1 δ i ( m + 1 ) ( 1 − ∣ w ∣ 2 ) m + 1 ,

has a unique solution c 1 i , c 2 i , c 3 i , and c 4 i , which is independent of w , since the determinant of the system

1 1 1 1 1 2 3 4 m ! ( m + 1 ) ! ( m + 2 ) ! 2 ( m + 3 ) ! 6 ( m + 1 ) ! ( m + 2 ) ! ( m + 3 ) ! 2 ( m + 4 ) ! 6 = 1 12 m ! ( m + 1 ) ! m 2 ( m − 1 ) ( m + 1 ) ≠ 0 .

For such c j i , j ∈ { 1 , 2 , 3 , 4 } , the function

g i , w ( z ) ≔ ∑ j = 1 4 c j i f j , w ( z )

satisfies the desired result.□

By a similar argument, we can obtain the following lemma.

Lemma 3

For any w ∈ D \ { 0 } and i , k ∈ { 0 , 1 , 2 } , there exists a function h i , w ∈ B 1 such that

h i , w ( k ) ( z ) = w ¯ k δ i k ( 1 − ∣ w ∣ 2 ) k ,

where δ i k is the Kronecker delta.

In order to estimate the essential norm of T u , v , φ m : B 1 → ℬ μ , we need the following two lemmas. The first one characterizes the compactness in terms of sequential convergence, whose proof is similar to that of [15, Proposition 3.11], so we omit the details.

Lemma 4

Let m ∈ N , u , v ∈ H ( D ) , and φ ∈ S ( D ) . Then, the operator T u , v , φ m : B 1 → ℬ μ is compact if and only if for each bounded sequence, { f n } n ∈ N in B 1 converges to zero uniformly on compact subsets of D as n → ∞ , we have ‖ T u , v , φ m f n ‖ ℬ μ → 0 as n → ∞ .

Lemma 5

[8] Every bounded sequence in B 1 has a subsequence that converges uniformly in D ¯ to a function in B 1 .

3 Main results

In this section, we formulate our main results. For simplicity of the expressions, we write

A 1 ( z ) = ∣ u ( z ) φ ′ ( z ) ∣ , A m ( z ) = ∣ v ′ ( z ) ∣ , A m + 1 ( z ) = ∣ v ( z ) φ ′ ( z ) ∣ .

We first give several characterizations of the generalized Stević-Sharma operator T u , v , φ m : B 1 → ℬ μ to be bounded.

Theorem 1

Let u , v ∈ H ( D ) , φ ∈ S ( D ) , m ∈ N , m > 1 , and μ be a radial weight. Then, the following statements are equivalent.

  1. The operator T u , v , φ m : B 1 → ℬ μ is bounded.

  2. u ∈ ℬ μ ,

    ∑ j = 1 4 sup w ∈ D ‖ T u , v , φ m f j , w ‖ ℬ μ < ∞ ,

    and

    ∑ i ∈ { 1 , m , m + 1 } sup z ∈ D μ ( z ) A i ( z ) < ∞ ,

    where f j , w are defined in (1).

  3. u ∈ ℬ μ , and

    ∑ i ∈ { 1 , m , m + 1 } sup z ∈ D μ ( z ) A i ( z ) ( 1 − ∣ φ ( z ) ∣ 2 ) i < ∞ .

Proof

(i) ⇒ (ii). Suppose that T u , v , φ m : B 1 → ℬ μ is bounded. Taking f 0 ( z ) = 1 ∈ B 1 we obtain, T u , v , φ m f 0 = u ∈ ℬ μ , that is,

(2) sup z ∈ D μ ( z ) ∣ u ′ ( z ) ∣ < ∞ .

For each w ∈ D and j ∈ { 1 , 2 , 3 , 4 } , ‖ f j , w ‖ B 1 ≲ 1 and hence by the boundedness of T u , v , φ m we have ‖ T u , v , φ m f j , w ‖ ℬ μ < ∞ . Therefore,

∑ j = 1 4 sup w ∈ D ‖ T u , v , φ m f j , w ‖ ℬ μ < ∞ .

Taking f 1 ( z ) = z ∈ B 1 and using the boundedness of T u , v , φ m : B 1 → ℬ μ , we obtain

∞ > ‖ T u , v , φ m f 1 ‖ ℬ μ ≥ sup z ∈ D μ ( z ) ∣ ( T u , v , φ m f 1 ) ′ ( z ) ∣ = sup z ∈ D μ ( z ) ∣ u ′ ( z ) φ ( z ) + u ( z ) φ ′ ( z ) ∣ ≥ sup z ∈ D μ ( z ) ∣ u ( z ) φ ′ ( z ) ∣ − sup z ∈ D μ ( z ) ∣ u ′ ( z ) φ ( z ) ∣ ,

which along with (2) and the fact that ∣ φ ( z ) ∣ < 1 , it follows that

(3) sup z ∈ D μ ( z ) ∣ u ( z ) φ ′ ( z ) ∣ ≤ ‖ T u , v , φ m f 1 ‖ ℬ μ + sup z ∈ D μ ( z ) ∣ u ′ ( z ) ∣ < ∞ .

Applying the operator T u , v , φ m for f m ( z ) = z m ∈ B 1 yields

∞ > ‖ T u , v , φ m f m ‖ ℬ μ ≥ sup z ∈ D μ ( z ) ∣ ( T u , v , φ m f m ) ′ ( z ) ∣ = sup z ∈ D μ ( z ) ∣ u ′ ( z ) φ ( z ) m + m u ( z ) φ ′ ( z ) φ ( z ) m − 1 + m ! v ′ ( z ) ∣ .

Using (2), (3), the fact that ∣ φ ( z ) ∣ < 1 , and the triangle inequality, we obtain

(4) sup z ∈ D μ ( z ) ∣ v ′ ( z ) ∣ < ∞ .

By choosing f m + 1 ( z ) = z m + 1 ∈ B 1 , we conclude that

∞ > ‖ T u , v , φ m f m + 1 ‖ ℬ μ ≥ sup z ∈ D μ ( z ) ∣ ( T u , v , φ m f m + 1 ) ′ ( z ) ∣ = sup z ∈ D μ ( z ) ∣ u ′ ( z ) φ ( z ) m + 1 + ( m + 1 ) u ( z ) φ ′ ( z ) φ ( z ) m + ( m + 1 ) ! v ′ ( z ) φ ( z ) + ( m + 1 ) ! v ( z ) φ ′ ( z ) ∣ .

By using (2), (3), and (4), in the same manner, we obtain

(5) sup z ∈ D μ ( z ) ∣ v ( z ) φ ′ ( z ) ∣ < ∞ .

Combining (3), (4), and (5), we deduce that

∑ i ∈ { 1 , m , m + 1 } sup z ∈ D μ ( z ) A i ( z ) < ∞ .

(ii) ⇒ (iii). Assume that (ii) holds. By Lemma 2, for each i ∈ { 1 , m , m + 1 } and φ ( w ) ≠ 0 , there exist constants c 1 i , c 2 i , c 3 i , and c 4 i such that

(6) g i , φ ( w ) ( z ) = ∑ j = 1 4 c j i f j , φ ( w ) ( z ) ∈ B 1 ,

and

g i , φ ( w ) ( k ) ( w ) = φ ( w ) ¯ i δ i k ( 1 − ∣ φ ( w ) ∣ 2 ) k ,

where f j , w are defined in (1) and k ∈ { 0 , 1 , m , m + 1 } . Then,

(7) ∞ > ∑ j = 1 4 sup w ∈ D ‖ T u , v , φ m f j , φ ( w ) ‖ ℬ μ ≳ sup w ∈ D ‖ T u , v , φ m g i , φ ( w ) ‖ ℬ μ ≥ μ ( w ) ∣ ( T u , v , φ m g i , φ ( w ) ) ′ ( w ) ∣ = μ ( w ) A i ( w ) ∣ φ ( w ) ∣ i ( 1 − ∣ φ ( w ) ∣ 2 ) i .

From (7) and (ii), for each i ∈ { 1 , m , m + 1 } , we have

sup ∣ φ ( w ) ∣ > 1 2 μ ( w ) A i ( w ) ( 1 − ∣ φ ( w ) ∣ 2 ) i < ∞

and

sup ∣ φ ( w ) ∣ ≤ 1 2 μ ( w ) A i ( w ) ( 1 − ∣ φ ( w ) ∣ 2 ) i ≲ sup w ∈ D μ ( w ) A i ( w ) < ∞ .

Therefore,

∑ i ∈ { 1 , m , m + 1 } sup z ∈ D μ ( z ) A i ( z ) ( 1 − ∣ φ ( z ) ∣ 2 ) i < ∞ .

(iii) ⇒ (i). Suppose that (iii) holds. For any f ∈ B 1 , by Lemma 1, we have

μ ( z ) ∣ ( T u , v , φ m f ) ′ ( z ) ∣ ≤ μ ( z ) ∣ u ′ ( z ) ∣ ∣ f ( φ ( z ) ) ∣ + ∑ i ∈ { 1 , m , m + 1 } μ ( z ) A i ( z ) ∣ f ( i ) ( φ ( z ) ) ∣ ≲ ‖ u ‖ ℬ μ + ∑ i ∈ { 1 , m , m + 1 } μ ( z ) A i ( z ) ( 1 − ∣ φ ( z ) ∣ 2 ) i ‖ f ‖ B 1 .

Moreover,

∣ ( T u , v , φ m f ) ( 0 ) ∣ = ∣ u ( 0 ) f ( φ ( 0 ) ) + v ( 0 ) f m ( φ ( 0 ) ) ∣ ≲ ∣ u ( 0 ) ∣ + ∣ v ( 0 ) ∣ ( 1 − ∣ φ ( 0 ) ∣ 2 ) m ‖ f ‖ B 1 .

Thus, T u , v , φ m : B 1 → ℬ μ is bounded. The proof is completed.□

By using Lemma 3 instead of Lemma 2, the following result may be proved in much the same way as Theorem 1.

Theorem 2

Let u , v ∈ H ( D ) , φ ∈ S ( D ) , and μ be a radial weight. Then, the following statements are equivalent.

  1. The operator T u , v , φ : B 1 → ℬ μ is bounded.

  2. u ∈ ℬ μ ,

    ∑ j = 1 3 sup w ∈ D ‖ T u , v , φ f j , w ‖ ℬ μ < ∞ ,

    and

    sup z ∈ D μ ( z ) ∣ u ( z ) φ ′ ( z ) + v ′ ( z ) ∣ + sup z ∈ D μ ( z ) ∣ v ( z ) φ ′ ( z ) ∣ < ∞ .

  3. u ∈ ℬ μ , and

    sup z ∈ D μ ( z ) ∣ u ( z ) φ ′ ( z ) + v ′ ( z ) ∣ 1 − ∣ φ ( z ) ∣ 2 + sup z ∈ D μ ( z ) ∣ v ( z ) φ ′ ( z ) ∣ ( 1 − ∣ φ ( z ) ∣ 2 ) 2 < ∞ .

Now, we estimate the essential norm of T u , v , φ m acting from the minimal Möbius invariant space to the Bloch-type space. Then, we obtain some equivalence conditions for compactness of T u , v , φ m .

Theorem 3

Let u , v ∈ H ( D ) , φ ∈ S ( D ) , m ∈ N , m > 1 , and μ be a radial weight such that T u , v , φ m : B 1 → ℬ μ is bounded. Then,

‖ T u , v , φ m ‖ e , B 1 → ℬ μ ≈ ∑ j = 1 4 limsup ∣ w ∣ → 1 ‖ T u , v , φ m f j , w ‖ ℬ μ ≈ ∑ i ∈ { 1 , m , m + 1 } limsup ∣ φ ( z ) ∣ → 1 μ ( z ) A i ( z ) ( 1 − ∣ φ ( z ) ∣ 2 ) i ,

where f j , w are defined in (1).

Proof

We first show that

‖ T u , v , φ m ‖ e , B 1 → ℬ μ ≳ ∑ j = 1 4 limsup ∣ w ∣ → 1 ‖ T u , v , φ m f j , w ‖ ℬ μ .

It is obvious that for each j ∈ { 1 , 2 , 3 , 4 } and w ∈ D , ‖ f j , w ‖ B 1 ≲ 1 . Moreover, f j , w converge to zero uniformly on compact subsets of D . For any compact operator K from B 1 into ℬ μ , by using some standard arguments (see, e.g., [34,35]), we obtain

lim ∣ w ∣ → 1 ‖ K f j , w ‖ ℬ μ = 0 .

It follows that

‖ T u , v , φ m − K ‖ B 1 → ℬ μ ≳ limsup ∣ w ∣ → 1 ‖ ( T u , v , φ m − K ) f j , w ‖ ℬ μ ≥ limsup ∣ w ∣ → 1 ‖ T u , v , φ m f j , w ‖ ℬ μ − limsup ∣ w ∣ → 1 ‖ K f j , w ‖ ℬ μ .

Therefore,

(8) ‖ T u , v , φ m ‖ e , B 1 → ℬ μ = inf K ‖ T u , v , φ m − K ‖ B 1 → ℬ μ ≳ ∑ j = 1 4 limsup ∣ w ∣ → 1 ‖ T u , v , φ m f j , w ‖ ℬ μ .

Next, we prove that

‖ T u , v , φ m ‖ e , B 1 → ℬ μ ≳ ∑ i ∈ { 1 , m , m + 1 } limsup ∣ φ ( z ) ∣ → 1 μ ( z ) A i ( z ) ( 1 − ∣ φ ( z ) ∣ 2 ) i .

Let { z j } be a sequence in D such that ∣ φ ( z j ) ∣ → 1 as j → ∞ . Since T u , v , φ m : B 1 → ℬ μ is bounded, for any compact operator K : B 1 → ℬ μ and i ∈ { 1 , m , m + 1 } , applying Lemma 4 and (7), we obtain

‖ T u , v , φ m − K ‖ B 1 → ℬ μ ≳ limsup j → ∞ ‖ T u , v , φ m g i , φ ( z j ) ‖ ℬ μ − limsup j → ∞ ‖ K g i , φ ( z j ) ‖ ℬ μ ≳ limsup j → ∞ μ ( z j ) A i ( z j ) ∣ φ ( z j ) ∣ i ( 1 − ∣ φ ( z j ) ∣ 2 ) i ,

where g i , φ ( z j ) are defined in (6). Therefore,

‖ T u , v , φ m ‖ e , B 1 → ℬ μ ≳ limsup j → ∞ μ ( z j ) A i ( z j ) ∣ φ ( z j ) ∣ i ( 1 − ∣ φ ( z j ) ∣ 2 ) i = limsup ∣ φ ( z ) ∣ → 1 μ ( z ) A i ( z ) ( 1 − ∣ φ ( z ) ∣ 2 ) i ,

from which we have

(9) ‖ T u , v , φ m ‖ e , B 1 → ℬ μ ≳ ∑ i ∈ { 1 , m , m + 1 } limsup ∣ φ ( z ) ∣ → 1 μ ( z ) A i ( z ) ( 1 − ∣ φ ( z ) ∣ 2 ) i .

Combining (8) and (9) yields

‖ T u , v , φ m ‖ e , B 1 → ℬ μ ≳ min ∑ j = 1 4 limsup ∣ w ∣ → 1 ‖ T u , v , φ m f j , w ‖ ℬ μ , ∑ i ∈ { 1 , m , m + 1 } limsup ∣ φ ( z ) ∣ → 1 μ ( z ) A i ( z ) ( 1 − ∣ φ ( z ) ∣ 2 ) i .

It is sufficient to show that

‖ T u , v , φ m ‖ e , B 1 → ℬ μ ≲ min ∑ j = 1 4 limsup ∣ w ∣ → 1 ‖ T u , v , φ m f j , w ‖ ℬ μ , ∑ i ∈ { 1 , m , m + 1 } limsup ∣ φ ( z ) ∣ → 1 μ ( z ) A i ( z ) ( 1 − ∣ φ ( z ) ∣ 2 ) i .

Define K r f ( z ) = f r ( z ) = f ( r z ) , where 0 ≤ r < 1 . Then, K r : B 1 → B 1 is a compact operator with ‖ K r ‖ ≤ 1 and f r → f uniformly on compact subsets of D as r → 1 clearly. Let { r j } ⊂ ( 0 , 1 ) be a sequence such that r j → 1 as j → ∞ . Then, for each j ∈ N , T u , v , φ m K r j : B 1 → ℬ μ is compact, and so

‖ T u , v , φ m ‖ e , B 1 → ℬ μ ≤ limsup j → ∞ ‖ T u , v , φ m − T u , v , φ m K r j ‖ B 1 → ℬ μ .

Therefore, we only need to show that

(10) limsup j → ∞ ‖ T u , v , φ m − T u , v , φ m K r j ‖ B 1 → ℬ μ ≲ min ∑ j = 1 4 limsup ∣ w ∣ → 1 ‖ T u , v , φ m f j , w ‖ ℬ μ , ∑ i ∈ { 1 , m , m + 1 } limsup ∣ φ ( z ) ∣ → 1 μ ( z ) A i ( z ) ( 1 − ∣ φ ( z ) ∣ 2 ) i .

For every f ∈ B 1 such that ‖ f ‖ B 1 ≤ 1 , we have

(11) ‖ ( T u , v , φ m − T u , v , φ m K r j ) f ‖ ℬ μ = ∣ T u , v , φ m f ( 0 ) − T u , v , φ m f r j ( 0 ) ∣ + sup z ∈ D μ ( z ) ∣ ( T u , v , φ m f − T u , v , φ m f r j ) ′ ( z ) ∣ ≤ ∣ ( f − f r j ) ( φ ( 0 ) ) u ( 0 ) ∣ + ∣ ( f − f r j ) ( m ) ( φ ( 0 ) ) v ( 0 ) ∣ ︸ E 0 + sup z ∈ D μ ( z ) ∣ ( f − f r j ) ( φ ( z ) ) u ′ ( z ) ∣ ︸ E 1 + sup ∣ φ ( z ) ∣ ≤ r N μ ( z ) ∑ i ∈ { 1 , m , m + 1 } ∣ ( f − f r j ) ( i ) ( φ ( z ) ) ∣ A i ( z ) ︸ E 2 + sup ∣ φ ( z ) ∣ > r N μ ( z ) ∑ i ∈ { 1 , m , m + 1 } ∣ ( f − f r j ) ( i ) ( φ ( z ) ) ∣ A i ( z ) ︸ E 3 ,

where N ∈ N such that r j ≥ 2 3 for all j ≥ N . Furthermore, we have ( f − f r j ) ( t ) → 0 uniformly on compact subsets of D as j → ∞ for any nonnegative integer t . Now, Theorem 1 implies

(12) limsup j → ∞ E 0 = limsup j → ∞ E 2 = 0 .

From Lemma 5,

(13) lim j → ∞ E 1 ≤ ‖ u ‖ ℬ μ lim j → ∞ sup z ∈ D ∣ ( f − f r j ) ( z ) ∣ = 0 .

Finally, we estimate E 3 .

(14) E 3 ≤ ∑ i ∈ { 1 , m , m + 1 } sup ∣ φ ( z ) ∣ > r N μ ( z ) ∣ f ( i ) ( φ ( z ) ) ∣ A i ( z ) ︸ F i + ∑ i ∈ { 1 , m , m + 1 } sup ∣ φ ( z ) ∣ > r N μ ( z ) ∣ r j i f ( i ) ( r j φ ( z ) ) ∣ A i ( z ) ︸ G i .

For each i ∈ { 1 , m , m + 1 } , using Lemma 1, (6), and (7), we obtain

(15) F i = sup ∣ φ ( z ) ∣ > r N ( 1 − ∣ φ ( z ) ∣ 2 ) i ∣ f ( i ) ( φ ( z ) ) ∣ ∣ φ ( z ) ∣ i μ ( z ) A i ( z ) ∣ φ ( z ) ∣ i ( 1 − ∣ φ ( z ) ∣ 2 ) i ≲ ‖ f ‖ B 1 sup ∣ φ ( z ) ∣ > r N ‖ T u , v , φ m g i , φ ( z ) ‖ ℬ μ ≲ ∑ j = 1 4 sup ∣ w ∣ > r N ‖ T u , v , φ m f j , w ‖ ℬ μ ,

and

(16) F i = sup ∣ φ ( z ) ∣ > r N ( 1 − ∣ φ ( z ) ∣ 2 ) i ∣ f ( i ) ( φ ( z ) ) ∣ μ ( z ) A i ( z ) ( 1 − ∣ φ ( z ) ∣ 2 ) i ≲ ‖ f ‖ B 1 sup ∣ φ ( z ) ∣ > r N μ ( z ) A i ( z ) ( 1 − ∣ φ ( z ) ∣ 2 ) i .

Taking the limits as N → ∞ in (15) and (16), we obtain

(17) limsup j → ∞ F i ≲ ∑ j = 1 4 limsup ∣ w ∣ → 1 ‖ T u , v , φ m f j , w ‖ ℬ μ

and

(18) limsup j → ∞ F i ≲ limsup ∣ φ ( z ) ∣ → 1 μ ( z ) A i ( z ) ( 1 − ∣ φ ( z ) ∣ 2 ) i .

Similarly, we have

(19) limsup j → ∞ G i ≲ ∑ j = 1 4 limsup ∣ w ∣ → 1 ‖ T u , v , φ m f j , w ‖ ℬ μ and limsup j → ∞ G i ≲ limsup ∣ φ ( z ) ∣ → 1 μ ( z ) A i ( z ) ( 1 − ∣ φ ( z ) ∣ 2 ) i .

Therefore, by (11)–(14) and (17)–(19), we obtain

limsup j → ∞ ‖ T u , v , φ m − T u , v , φ m K r j ‖ B 1 → ℬ μ = limsup j → ∞ sup ‖ f ‖ B 1 ≤ 1 ‖ ( T u , v , φ m − T u , v , φ m K r j ) f ‖ ℬ μ ≲ ∑ j = 1 4 limsup ∣ w ∣ → 1 ‖ T u , v , φ m f j , w ‖ ℬ μ ,

and

limsup j → ∞ ‖ T u , v , φ m − T u , v , φ m K r j ‖ B 1 → ℬ μ ≲ ∑ i ∈ { 1 , m , m + 1 } limsup ∣ φ ( z ) ∣ → 1 μ ( z ) A i ( z ) ( 1 − ∣ φ ( z ) ∣ 2 ) i .

From the last two inequalities, we obtain (10) and the proof is completed.□

Corollary 1

Let u , v ∈ H ( D ) , φ ∈ S ( D ) , m ∈ N , m > 1 , and μ be a radial weight. Suppose that T u , v , φ m : B 1 → ℬ μ is bounded, then the following statements are equivalent.

  1. The operator T u , v , φ m : B 1 → ℬ μ is compact.

  2. ∑ j = 1 4 limsup ∣ w ∣ → 1 ‖ T u , v , φ m f j , w ‖ ℬ μ = 0 .

  3. ∑ i ∈ { 1 , m , m + 1 } limsup ∣ φ ( z ) ∣ → 1 μ ( z ) A i ( z ) ( 1 − ∣ φ ( z ) ∣ 2 ) i = 0 .

By the same method as in the proof of Theorem 3, we can obtain the following results for the case m = 1 , namely, the Stević-Sharma operator.

Theorem 4

Let u , v ∈ H ( D ) , φ ∈ S ( D ) , and μ be a radial weight such that T u , v , φ : B 1 → ℬ μ is bounded. Then,

‖ T u , v , φ ‖ e , B 1 → ℬ μ ≈ ∑ j = 1 3 limsup ∣ w ∣ → 1 ‖ T u , v , φ f j , w ‖ ℬ μ ≈ limsup ∣ φ ( z ) ∣ → 1 μ ( z ) ∣ u ( z ) φ ′ ( z ) + v ′ ( z ) ∣ 1 − ∣ φ ( z ) ∣ 2 + limsup ∣ φ ( z ) ∣ → 1 μ ( z ) ∣ v ( z ) φ ′ ( z ) ∣ ( 1 − ∣ φ ( z ) ∣ 2 ) 2 .

Corollary 2

Let u , v ∈ H ( D ) , φ ∈ S ( D ) , and μ be a radial weight. Suppose that T u , v , φ : B 1 → ℬ μ is bounded, then the following statements are equivalent.

  1. The operator T u , v , φ : B 1 → ℬ μ is compact.

  2. ∑ j = 1 3 limsup ∣ w ∣ → 1 ‖ T u , v , φ f j , w ‖ ℬ μ = 0 .

  3. limsup ∣ φ ( z ) ∣ → 1 μ ( z ) ∣ u ( z ) φ ′ ( z ) + v ′ ( z ) ∣ 1 − ∣ φ ( z ) ∣ 2 + limsup ∣ φ ( z ) ∣ → 1 μ ( z ) ∣ v ( z ) φ ′ ( z ) ∣ ( 1 − ∣ φ ( z ) ∣ 2 ) 2 = 0 .

Acknowledgements

The author is grateful to the referees and the editor for bringing important references to our attention and many valuable suggestions that greatly improved the final version of this manuscript.

  1. Funding information: This work was supported by the National Natural Science Foundation of China (No. 12101188).

  2. Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The author states no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzes during this study.

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Received: 2022-04-08
Revised: 2023-02-03
Accepted: 2023-05-17
Published Online: 2023-10-03

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2022-0245
Keywords for this article
generalized Stević-Sharma operator; minimal Möbius invariant space; Bloch-type space; essential norm
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  51. Asymptotic stability of equilibria for difference equations via fixed points of enriched Prešić operators
  52. Asymptotic behavior of resolvents of equilibrium problems on complete geodesic spaces
  53. A system of additive functional equations in complex Banach algebras
  54. New inertial forward–backward algorithm for convex minimization with applications
  55. Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem with the p-Laplacian operator
  56. Analysis of Cauchy problem with fractal-fractional differential operators
  57. Common best proximity points for a pair of mappings with certain dominating property
  58. Investigation of hybrid fractional q-integro-difference equations supplemented with nonlocal q-integral boundary conditions
  59. The structure of fuzzy fractals generated by an orbital fuzzy iterated function system
  60. On the structure of self-affine Jordan arcs in ℝ2
  61. Solvability for a system of Hadamard-type hybrid fractional differential inclusions
  62. Three solutions for discrete anisotropic Kirchhoff-type problems
  63. On split generalized equilibrium problem with multiple output sets and common fixed points problem
  64. Special Issue on Computational and Numerical Methods for Special Functions - Part II
  65. Sandwich-type results regarding Riemann-Liouville fractional integral of q-hypergeometric function
  66. Certain aspects of Nörlund ℐ-statistical convergence of sequences in neutrosophic normed spaces
  67. On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditions
  68. Some identities on generalized harmonic numbers and generalized harmonic functions
  69. Study of degenerate derangement polynomials by λ-umbral calculus
  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
  72. Compositions of positive integers with 2s and 3s
  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
  77. On the order of approximation by modified summation-integral-type operators based on two parameters
  78. Bernstein-type operators on elliptic domain and their interpolation properties
  79. A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities
  80. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part II
  81. Application of fractional quantum calculus on coupled hybrid differential systems within the sequential Caputo fractional q-derivatives
  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
  86. The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method
  87. Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves
  88. Petri net analysis of a queueing inventory system with orbital search by the server
  89. Asymptotic stability of an epidemiological fractional reaction-diffusion model
  90. On the stability of a strongly stabilizing control for degenerate systems in Hilbert spaces
  91. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part I
  92. New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
  95. Some results on fractional Hahn difference boundary value problems
  96. Properties of a subclass of analytic functions defined by Riemann-Liouville fractional integral applied to convolution product of multiplier transformation and Ruscheweyh derivative
  97. Special Issue on Development of Fuzzy Sets and Their Extensions - Part I
  98. The cross-border e-commerce platform selection based on the probabilistic dual hesitant fuzzy generalized dice similarity measures
  99. Comparison of fuzzy and crisp decision matrices: An evaluation on PROBID and sPROBID multi-criteria decision-making methods
  100. Rejection and symmetric difference of bipolar picture fuzzy graph
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