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

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


Introduction
Let be the open unit disk in the complex plane and the set of positive integers.Denote by H ( ) the class of all analytic functions on and S ( ) the family of all analytic self-maps of .
The set of all conformal automorphisms of forms a group, called the Möbius group, and is denoted by Aut( ).It is well known from complex analysis that every element of Aut( ) has the form e σ z iθ w ( ), where θ is a real number and is a special automorphism of exchanging the points w and 0. Let X be a linear space of analytic functions on .Then, X is said to be Möbius invariant if for all ∈ f X and ∈ ν Aut( ), ∘ ∈ f ν X and satisfies that ∘ = f ν f X X ‖ ‖ ‖ ‖ (see [1]).A typical example of Möbius invariant space is the analytic Besov space B p .Recall that for < < ∞ p 1 , a function ∈ f H ( ) belongs to B p if where dA is the normalized Lebesgue area measure on .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 ( ), which have a representation as: for some sequences in .The norm in B 1 is defined by: 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 ( ) for which ″ ∈ f L A , d 1 ( ), and there exist constants C 1 and C 2 such that For more studies of B 1 space, see also [3][4][5][6][7][8].Suppose that μ is a weight, namely, a strictly positive continuous function on .We also assume that μ is radial: | | , the induced space 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][11][12][13][14].
Suppose that ∈ φ S ( ) and ∈ u H ( ), the composition and multiplication operators on H ( ) are defined, respectively, by: where ∈ f H ( ) and ∈ z .The product of these two operators is known as the weighted composition operator 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 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 DC φ and C D φ (see, for example, [11,[16][17][18][19]).In [20,21], Stević and co-workers introduced the so-called Stević-Sharma operator as follows: where ∈ u v H , ( ) and ∈ φ S ( ).By taking some specific choices of the involving symbols, we can easily obtain the general product-type operators: 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: and studied its boundedness, compactness, and essential norm from Hardy space into the nth 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][11][12][13][14][26][27][28][29][30][31][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 μ .As a corollary, we give the characterizations of its compactness.
Recall that the essential norm of a bounded linear operator where X and Y are the Banach spaces.Note that =

‖ ‖ 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 CY .Moreover, we write

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]).
For any ∈ w and ∈ j , set It is easily seen that , converges to 0 uniformly on compact subsets of as where δ ik is the Kronecker delta.
Proof.For any ∈ ⧹ w 0 { } and constants c c c , , 1 2 3 , and c 4 , let has a unique solution c c c , , , and c i 4 , which is independent of w, since the determinant of the system Generalized Stević-Sharma operators  3 satisfies the desired result.□ By a similar argument, we can obtain the following lemma.
where δ ik is the Kronecker delta.
In order to estimate the essential norm of → T B : , 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.
, ( ), and ∈ φ S ( ).Then, the operator → T B : is compact if and only if for each bounded sequence, in B 1 converges to zero uniformly on compact subsets of as → ∞ n , we have Lemma 5.
[8] Every bounded sequence in B 1 has a subsequence that converges uniformly in to a function in B 1 .

Main results
In this section, we formulate our main results.For simplicity of the expressions, we write We first give several characterizations of the generalized Stević-Sharma operator → T B : to be bounded.
and μ be a radial weight.Then, the following statements are equivalent.
(i) The operator → T B : where f j w , are defined in (1). (iii) ∈ u μ , and For each ∈ w and ‖ ‖ and hence by the boundedness of ( ) and using the boundedness of → T B : , we obtain which along with (2) and the fact that , it follows that Applying the operator , and the triangle inequality, we obtain By choosing ( ) , we conclude that By using (2), (3), and (4), in the same manner, we obtain Combining ( 3), (4), and (5), we deduce that and where f j w , are defined in (1) and From ( 7) and (ii), for each { } (iii) ⇒ (i).Suppose that (iii) holds.For any ∈ f B 1 , by Lemma 1, we have 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.
and μ be a radial weight.Then, the following statements are equivalent.(i) The operator → T B : Now, we estimate the essential norm of T u v φ m , , acting from the minimal Möbius invariant space to the Blochtype space.Then, we obtain some equivalence conditions for compactness of T u v φ m , , .
Proof.We first show that , , , , converge to zero uniformly on compact subsets of .For any compact operator K from B 1 into μ , by using some standard arguments (see, e.g., [34,35]), we obtain Next, we prove that is bounded, for any compact operator → K B : } , applying Lemma 4 and ( 7), we obtain where g i φ z , j ( ) are defined in (6).Therefore, from which we have Combining ( 8) and ( 9) yields is compact, and so Therefore, we only need to show that For every , 1 where uniformly on compact subsets of as → ∞ j for any nonnegative integer t.Now, Theorem 1 implies From Lemma 5, Finally, we estimate E 3 .

m 1 ,
From the last two inequalities, we obtain(10) and the proof is completed.and μ be a radial weight.Suppose that → 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.