New characterizations for the essential norms of generalized weighted composition operators between Zygmund type spaces

We give different types of new characterizations for the boundedness and essential norms of generalized weighted composition operators between Zygmund type spaces. Consequently, we obtain new characterizations for the compactness of such operators.


Introduction
Let For a weight ν, the associated weight ν is defined by It is known that for the standard weights ν α (z) = (1 − |z| 2 ) α , 0 < α < ∞, and for the logarithmic weight ν log (z) = log 2 1−|z| 2 −1 , the associated weights and weights are the same.
For each 0 < α < ∞, the Bloch type space B α consists of all functions f ∈ H(D) for which The space B α is a Banach space equipped with the norm |f (e i(θ+h) ) + f (e i(θ−h) ) − 2f (e iθ )| h < ∞, where the supremum is taken over all θ ∈ R and h > 0. By [2,Theorem 5.3], an analytic function f belongs to Z if and only if sup z∈D (1 − |z| 2 )|f ′′ (z)| < ∞. Motivated by this, for each 0 < α < ∞, the Zygmund type space Z α is defined to be the space of all functions f ∈ H(D) for which The space Z α is a Banach space equipped with the norm for each f ∈ Z α . The little Zygmund type space Z α,0 is the closed subspace of Z α consists of those functions f ∈ Z α satisfying lim |z|→1 (1 − |z| 2 ) α |f ′′ (z)| = 0.
Recall that for the Banach spaces X and Y , the space of all bounded operators T : X → Y is denoted by B(X, Y ) and the operator norm of T ∈ B(X, Y ) is denoted by T X→Y . The closed subspace of B(X, Y ) containing all compact operators T : X → Y is denoted by K(X, Y ). The essential norm of T ∈ B(X, Y ), denoted by T e,X→Y , is defined as the distance from T to K(X, Y ), that is T e,X→Y = inf{ T − K X→Y : K ∈ K(X, Y )}.
Clearly, an operator T ∈ B(X, Y ) is compact if and only if T e,X→Y = 0. Therefore, essential norm estimates of bounded operators result in necessary and/or sufficient conditions for the compactness of such operators. Essential norm estimates of different types of operators between various classes of Banach spaces have been studied by many authors. See, for example, [3,4,5,6,13,16] and references therein.
Let u and ϕ be analytic functions on D such that ϕ(D) ⊆ D. The weighted composition operator uC ϕ is defined by uC ϕ f = u · f • ϕ for all f ∈ H(D). When u = 1 we get the well-known composition operator C ϕ given by C ϕ f = f • ϕ for all f ∈ H(D). Weighted composition operators appear in the study of dynamical systems and also it is known that isometries on many analytic function spaces are of the canonical forms of weighted composition operators. Operator theoretic properties of (weighted) composition operators have been studied by many authors between different classes of analytic function spaces. See, for example, [5,6,11,12] and the references therein.
For each non-negative integer k, the generalized weighted composition operator D k ϕ,u is defined by for each f ∈ H(D) and z ∈ D. The class of generalized weighted composition operators include weighted composition operators uC ϕ = D 0 ϕ,u , composition operators followed by differentiation DC ϕ = D 1 ϕ,ϕ ′ and composition operators proceeded by differentiation C ϕ D = D 1 ϕ,1 [9]. Also, weighted types of operators DC ϕ and C ϕ D are of the form D k ϕ,u , that is uDC ϕ = D 1 ϕ,uϕ ′ and uC ϕ D = D 1 ϕ,u [10]. Boundedness and compactness of generalized weighted composition operators have been studied between Bloch type spaces and Zygmund type spaces in [3,8], and between Bloch type spaces and weighted-type spaces in [7,17]. Essential norms of generalized weighted composition operators in these cases have been studied in [3,4,13]. In [15], different characterizations for the boundedness and compactness of these operators between Bloch type spaces are given and also their essential norms are investigated in [16]. In this paper we first study boundedness of generalized weighted composition operators between Zygmund type spaces and give new characterizations for the boundedness of these operators. Then, we find estimates for the essential norms of such operators in terms of the new characterizations. Consequently, we obtain different types of characterizations for the compactness of such operators.
The following lemma, which will be used in the next chapters, collects some useful estimates for the functions in Zygmund type spaces. See, for example, [13] and references therein.
It is known that for each n ≥ 2 and 0 < α < ∞ we have for all f ∈ B α and z ∈ D, see [18]. Therefore, for each n ≥ 2 and 0 < α < ∞ we have for all f ∈ Z α and z ∈ D. Note that, by the definition of Zygmund type spaces, it is clear that (1.1) also holds in the case of n = 1.
In this paper, for real scalars A and B, the notation A B means A ≤ cB for some positive constant c. Also, the notation A ≈ B means A B and B A.

Boundedness
For each a ∈ D, the following test functions in H(D) will be used in our proofs Moreover, in order to simplify the notations, we define In the next theorem, we give three different characterizations for the boundedness of D n ϕ,u : Z α → Z β .
Moreover, this is also equivalent to in the special case of n = 1 and 0 < α < 1.
Proof. (i) ⇒ (ii) Suppose that n ≥ 1 and D n ϕ,u : Z α → Z β is a bounded operator. Since {j α−1 I j } is a bounded sequence in B α , see for example [14], one can see that {j α−2 I j+1 } is a bounded sequence in Z α . Therefore, boundedness of D n ϕ,u : Recalling the definition of f a , g a and h a , one can see that for each a, z ∈ D. Note that Also, by Stirling's formula, we know that Γ(j+α) j!Γ(α) ≈ j α−1 as j → ∞, see [3]. Consequently, and since a ∈ D was arbitrary, we get sup a∈D D n ϕ,u f a Z β < ∞. Applying a similar argument to the functions g a and h a implies (iii).
Then, one can see that k a ∈ Z α , sup a∈D k a Zα < ∞, k and by the definition of · Z β we have Therefore, On the other hand, In order to prove that B(u, ϕ, α, β, n) < ∞, the similar approach can be applied using the test functions l a defined as Also, for the proof of A(u, ϕ, α, β, n) < ∞ the argument is similar, using the test functions The special case of n = 1 and 0 < α < 1 can be proved by a similar argument and applying Lemma 1.1.
In the case of n = 1 and α = 1 we have the following result.
Proof. First note that, as mentioned in the proof of Theorem 2.1, {j α−2 I j+1 } is a bounded sequence in Z α . Since polynomials are dense in Z α,0 , by [1, Proposition 2.1], {j α−2 I j+1 } converges weakly to 0 in Z α,0 and hence in Z α . Therefore, for each compact operator K : Z α → Z β , we have lim j→∞ K(j α−2 I j+1 ) Z β = 0 and consequently D n ϕ,u e,Zα→Z β = inf Next we prove that D n ϕ,u e,Zα→Z β ≈ max{E, F, G}. One can see that f a , g a , h a ∈ Z α,0 having uniformly bounded · Zα norm. Also, these functions converge to 0 uniformly on compact subsets of D, as |a| → 1, which implies their weak convergence to 0 in Z α . Therefore, for each compact operator K : The same argument is valid for g a and h a , that is Since K : Z α → Z β was an arbitrary compact operator, we get For each 0 < r < 1 and f ∈ H(D) the function f r ∈ H(D) is defined by f r (z) = f (rz) for all z ∈ D. Note that f r → f uniformly on compact subsets of D when r → 1. Also, the operator K r : Z α → Z α given by K r f = f r is a compact operator for each 0 < r < 1. Let {r j } be a sequence in (0, 1) converging to 1 as j → ∞. Then, D n ϕ,u K r j : Z α → Z β is a compact operator for each j ≥ 1. Hence, Therefore, it is enough to prove that lim sup j→∞ D n ϕ,u − D n ϕ,u K r j Zα→Z β max{E, F, G}.
Note that for every f ∈ Z α with f Zα ≤ 1 we have Since r j → 1, we have Therefore, for each N ≥ 1, we get lim sup respectively. Since the operator D n ϕ,u : Z α → Z β is bounded, by Theorem 2.1, we have Also, for each k ≥ 1, f (k) r j → f (k) uniformly on compact subsets of D, since f r j → f uniformly on compact subsets of D. These facts imply that S 1 = S 3 = S 5 = 0, and hence we just need to estimate S 2 , S 4 and S 6 . By applying (1.1), Lemma 1.1 and using the test functions m a defined in Theorem 2.1, we have Therefore, as N → ∞, we get Similarly, by applying the test functions k a and l a , one can prove S 4 max{E, F, G}, S 6 max{E, F, G}, and therefore By applying (3.2), we get and similarly one can see that S 4 B and S 6 C. Therefore, In order to prove the converse of (3.4), let {z j } be a sequence in D such that |ϕ(z j )| → 1, as j → ∞, and define the test functions where k a , l a and m a are defined in the proof of Theorem 2.1. Then, {k j }, {l j } and {m j } are bounded sequences in Z α,0 which converge to zero on compact subsets of D. Also, for each j ≥ 1 we have Thus, for any compact operator K : Z α → Z β we get Hence, To prove the final case, first note that by Theorem 2.1 we have Recall that, for each a, z ∈ D For each N ≥ 1, as in the proof of Theorem 2.1 we have Therefore, when N → ∞, we get A similar approach implies that This, along with (3.1), completes the proof.
By applying these facts, our results in this paper containing terms of the type ω(z) ν(ϕ(z)) |u(z)| can be restated in terms of u and ϕ n . See, for example, [12] and references therein for these types of results.