Essential Norms of Weighted Composition Operators from the α-Bloch Space to a Weighted-Type Space on the Unit Ball

This paper finds some lower and upper bounds for the essential norm of the weighted composition operator from -Bloch spaces to the weighted-type space on the unit ball for the case .


Introduction
Let B {z ∈ C n : |z| < 1} be the open unit ball in C n , H B be the class of all holomorphic functions on the unit ball, and let H ∞ B be the class of all bounded holomorphic functions on B with the norm f ∞ sup z∈B f z .
1 A positive continuous function φ on the interval 0, 1 is called normal see 1 if there is δ ∈ 0, 1 and a and b, 0 < a < b such that φ r 1 − r a is decreasing on δ, 1 and lim From now on, if we say that a function φ : B → 0, ∞ is normal, we will also assume that it is radial, that is, where μ is normal on the interval 0, 1 .For μ z 1 − |z| 2 β , β > 0, we obtain the weighted space H ∞ β H ∞ β B see, e.g., 2, 3 .The α-Bloch space B α B α B , α > 0, is the space of all f ∈ H B such that With the norm the space B α is a Banach space 4-6 .
The little α-Bloch space B α 0 Let u ∈ H B and ϕ be a holomorphic self-map of the unit ball.Weighted composition operator on H B , induced by u and ϕ is defined by This operator can be regarded as a generalization of a multiplication operator and a composition operator.It is interesting to provide a function theoretic characterization when u and ϕ induce a bounded or compact weighted composition operator between some spaces of holomorphic functions on B. For some classical results in the topic see, e.g., 5 .For some recent results on this and related operators, see, e.g., 2-4, 7-25 and the references therein.In 18 , Ohno has characterized the boundedness and compactness of weighted composition operators between H ∞ and the Bloch space B on the unit disk.In the setting of the unit polydisk D n , we have given some necessary and sufficient conditions for a weighted composition operator to be bounded or compact from H ∞ D n to the Bloch space B D n in 12 see, also 21 .Corresponding results for the case of the unit ball are given in 14 .Among other results, in 14 , we have given some necessary and sufficient conditions for the Stevo Stević 3 compactness of the operator uC ϕ : B α B → H ∞ B , which we incorporate in the following theorem.
Theorem A. Let ϕ ϕ 1 , . . ., ϕ n be a holomorphic self-map of B and u ∈ H B .
a If α > 1, then the following statements are equivalent: 1.9 b If α 1, then the following statements are equivalent: We would also like to point out that if α ≥ 1, then the boundedness of uC ϕ : B α → H ∞ μ and uC ϕ : B α 0 → H ∞ μ are equivalent see 22 for the case α 1, and the proof of Theorem 3 in 14 .
The essential norm of an operator is its distance in the operator norm from the compact operators.More precisely, assume that X 1 and X 2 are Banach spaces and A : X 1 → X 2 is a bounded linear operator, then the essential norm of A, denoted by A e,X 1 →X 2 , is defined as follows: where Since the set of all compact operators is a closed subset of the set of bounded operators, it follows that an operator A is compact if and only if A e,X 1 →X 2 0.
Motivated by Theorem A, in this paper, we find some lower and upper bounds for the essential norm of the weighted composition operator uC ϕ : Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to another.The notation a b means that there is a positive constant C such that a ≤ Cb.If both a b and b a hold, then we say that a b.

Auxiliary results
In this section, we quote several auxiliary results which we need in the proofs of the main results in this paper.The following lemma should be folklore.
for some C > 0 independent of f.
The proof of the lemma for the case α / 1 can be found, for example, in 26 .The formulation of the corresponding estimate in 26 , for the case α 1, is slightly different.In this case, Lemma 2.1 follows from the following estimate: The next lemma can be proved in a standard way see, e.g., the proofs of the corresponding results in 5, 27-29 .

2.4
Proof.We have from which it easily follows that Hence, the points are stationary for the function g s x .Since x M > 1, it follows that g s x attains its maximum on the interval 0, 1 at the point

2.10
By some long but elementary calculations, it follows that

Estimates of the essential norm of uC
In this section, we prove the main results in this paper.Before we formulate and prove these results, we prove another auxiliary result.
, and the operator uC ϕ : converging to zero on compacts of B as k → ∞.Then, we have as k → ∞, since ϕ B is contained in the ball |w| ≤ ϕ ∞ which is a compact subset of B, according to the assumption, ϕ ∞ < 1.Hence, by Lemma 2.2 , the operator uC ϕ :

3.2
for some positive constant C.
μ is compact which is equivalent with uC ϕ e,B α →H ∞ μ 0, and, consequently, uC ϕ e,B α 0 →H ∞ μ 0. On the other hand, it is clear that in this case the condition |ϕ z | → 1 is vacuous, so that inequalities in 3.2 are vacuously satisfied.
Hence, assume ϕ ∞ 1.Let z k k∈N be a sequence in B such that lim k→∞ |ϕ z k | 1 and ε ∈ 0, 1 be fixed.Set Then, by 6, Theorem 7.5 , it follows that f ε k converges to zero weakly as k → ∞.Hence, for every compact operator for every compact operator L : Letting k → ∞ in 3.4 and using the definition of f ε k , we obtain where C ε, α is the quantity in 2.12 .Taking in 3.5 the infimum over the set of all compact operators L : B α 0 → H ∞ μ , then letting ε → 0 in such obtained inequality, and using 2.13 , we obtain from which the first inequality in 3.2 follows.
Since the second inequality in 3.2 is obvious, we only have to prove the third one.By Lemma 3.1, we have that for each fixed ρ ∈ 0, 1 the operator uC ρϕ : B α → H ∞ μ is compact.Let δ ∈ 0, 1 be fixed, and let ρ m m∈N be a sequence of positive numbers which increasingly converges to 1, then for each fixed m ∈ N, we have

3.7
By the mean-value theorem, we have sup as m → ∞.

8
Abstract and Applied Analysis Moreover, by Lemma 2.1 case α > 1 , and known inequality where f r z f rz , r ∈ 0, 1 , we have On the other hand, for every compact operator L : B 0 → H ∞ μ , we have 3.17 Using 3.15 , letting k → ∞ in 3.17 , and applying 3.16 , it follows that

3.18
Taking in 3.18 the infimum over the set of all compact operators L : from which the first inequality in 3.12 follows.
As in Theorem 3.2, we need only to prove the third inequality in 3.12 .
Recall that for each ρ ∈ 0, 1 , the operator uC ρϕ : B → H ∞ μ is compact.Let the sequence ρ m m∈N be as in Theorem 3.2.Note that inequality 3.

1 −
r b is increasing on δ,

2 f − f ρ m B μ z u z ln 1 ϕ z 1 3 . 21 From 3 .Corollary 3 . 5 .
7 and relationship 3.8 also hold for α 1.Hence, we should only estimate the quantityI δ m sup |ϕ z |>δ sup f B ≤1 μ z u z f ϕ z − f ρ m ϕ z .3.20On the other hand, by Lemma 2.1 case α 1 applied to the function f − f ρ m , which belongs to the Bloch space for each m ∈ N, and inequality 3.9 with α 1, we haveμ z u z f ϕ z − f ρ m ϕ z ≤ 1 − ϕ z ≤ f B μ z u z ln 1 ϕ z 1 − ϕ z .21 ,by letting m → ∞ in 3.7 and using 3.8 with α 1 , and letting δ → 1 in such obtained inequality, we obtain uC ϕ e,B→H ∞ μ ≤ lim sup Assume u ∈ H B , μ is normal, ϕ ϕ 1 , . . ., ϕ n is a holomorphic self-map of B, and the operator uC ϕ : B or B α 0 → H ∞ μ is bounded.Then, uC ϕ : B or B α 0 → H ∞ μ is compact if and only if Replacing 3.10 in 3.7 , letting in such obtained inequality m → ∞, employing 3.8 , and then letting δ → 1, the third inequality in 3.2 follows, finishing the proof of the theorem.Assume α > 1, μ is normal, u ∈ H B , ϕ ϕ 1 , . .., ϕ n is a holomorphic self-map of B, and the operator uC ϕ : B α or B α 0 → H ∞ μ is bounded.Then, uC ϕ : B α or B α Assume ϕ z k k∈N is a sequence in B such that |ϕ z k | → 1 as k → ∞.Note that h ϕ z k k∈N is a bounded sequence in B moreover in B 0 converging to zero uniformly on compacts of B. Then, by 6, Theorem 7.5 , it follows that h ϕ z k converges to zero weakly as k → ∞.Hence, for every compact operator L : B 0 → H ∞