Weighted Composition Operators from Hardy to Zygmund Type Spaces

and Applied Analysis 3 By (14) and the boundedness of the function φ(z), we get 1 = sup z∈D (1 − |z| 2 ) α 󵄨 󵄨 󵄨 󵄨 󵄨 2φ 󸀠 (z) u 󸀠 (z) + φ 󸀠󸀠 (z) u (z) 󵄨 󵄨 󵄨 󵄨 󵄨 < +∞.

Let  be a fixed analytic function on the open unit disk.Define a linear operator   on the space of analytic functions on , called a weighted composition operator, by    =  ⋅ ( ∘ ), where  is an analytic function on .We can regard this operator as a generalization of a multiplication operator and a composition operator.In recent years the weighted composition operator has received much attention and appears in various settings in the literature.For example, it is known that isometries of many analytic function spaces are weighted composition operators (see [5], for instance).Their boundedness and compactness have been studied on various Banach spaces of analytic functions, such as Hardy, Bergman, BMOA, Bloch-type, and Zygmund spaces; see, for example, [6][7][8][9][10][11]. Also, it has been studied from one Banach space of analytic functions to another; one may see [12][13][14][15][16][17][18][19][20][21][22][23].
The purpose of this paper is to consider the weighted composition operators from the Hardy space   (0 <  < ∞) to the Zygmund type spaces Z  .Our main goal is to characterize boundedness and compactness of the operators   from   to Z  in terms of function theoretic properties of the symbols  and .
When  = 1, it is called the Zygmund space.From a theorem by Zygmund (see [26, And the polynomials are norm-dense in closed subspace Z ,0 .For some other information on this space and some operators on it, see, for example, [28][29][30][31].Throughout this paper, constants are denoted by , (), they are positive, and () are only depending on  and may differ from one occurrence to the another.

Auxiliary Results
In order to prove the main results of this paper.We need some auxiliary results.The first lemma is well known.
This is obvious.
Theorem 5. Let  > 0, 0 <  < ∞, and  be an analytic function on the unit disc  and  an analytic self-map of .

Compactness of 𝑢𝐶 𝜑
In order to prove the compactness of   from   to the Zygmund spaces Z  , we require the following lemmas.Lemma 6.Let  > 0, 0 <  < ∞, and  be an analytic function on the unit disc  and  an analytic self-map of .Suppose that   is a bounded operator from   to Z  .Then   is compact if and only if, for any bounded sequence {  } in   which converges to 0 uniformly on compact subsets of , one has ‖  (  )‖ * → 0 as  → ∞.
The proof is similar to that of Proposition 3.11 in [32].The details are omitted.Theorem 7. Let  > 0, 0 <  < ∞,  be an analytic function on the unit disc  and  an analytic self-map of .Then   is a compact operator from   to Z  if and only if   is a bounded operator and the following are satisfied: By a direct calculation we obtain that ℎ   0 ( → ∞) on compact subsets of  and sup  ‖ℎ  ‖  < ∞.Consequently, {ℎ  } is a bounded sequence in   which converges to 0 uniformly on compact subsets of .Then lim  → ∞ ‖  (ℎ  )‖ * = 0 by Lemma 6.Note that The proof of the necessary is completed.Conversely, Suppose that (37) hold.Since   is a bounded operator, from Theorem 4, we have Let {  } be a bounded sequence in   with ‖  ‖  ≤ 1 and   → 0 uniformly on compact subsets of .We only prove lim  → ∞ ‖  (  )‖ * = 0 by Lemma 6.By the assumption, for any  > 0, there is a constant , 0 <  < 1, such that  < |()| < 1 implies Hence   is compact.This completes the proof of Theorem 7.
In order to prove the compactness of   on the little Zygmund spaces Z ,0 , we require the following lemma.Lemma 8. Let  ⊂ Z ,0 .Then  is compact if and only if it is closed, bounded and satisfies The proof is similar to that of Lemma 1 in [1], but we omit it.
Theorem 9. Let  > 0, 0 <  < ∞,  be an analytic function on the unit disc  and  an analytic self-map of .Then   is compact from   to the little Zygmund type spaces Z ,0 if and only if (28), (29), and (30) hold.
Next, note that the proof of Theorem 4 and the fact that the functions given in (18)  (58) This completes the proof of Theorem 9.
Remark 10.From Theorems 5 and 9, we conjecture that   :   → Z ,0 is compact if and only if   :   → Z ,0 is bounded.
→ Z  is a bounded operator if and only if the following are satisfied: Corollary 12. Let  > 0, 0 <  < ∞, and  be an analytic self-map of .Then   :   → Z  is a compact operator if and only if   is bounded and the following are satisfied: Let  > 0, 0 <  < ∞, and  be an analytic self-map of .Then   :   → Z ,0 is a compact operator if and only ifIn the formulation of corollary, we use the notation   on () defined by    =  for  ∈ ().Taking () =  from Theorems 4, 5, 7, and 9, we obtain the following results about the characterization of the boundedness and compactness of pointwise multiplier   :   → Z  ( Z ,0 ).