Composition Operators from Certain μ-Bloch Spaces to Q P Spaces

and Applied Analysis 3


Introduction
Let D be the unit disc in the complex plane C and (D) the space of all analytic functions on D with the topology of uniform convergence on compact subsets of D. If  ∈ (D), we let   () = (), 0 <  < 1, be the dilation of .The  ∞ space consists of all functions  ∈ (D) satisfying sup ∈D |()| < ∞.The Bloch space B consists of all functions  ∈ (D) for which         B := sup B equipped with the norm ‖‖ := |(0)| + ‖‖ B becomes a Banach space (see [1,2]).For  > 0, the -Bloch space B  consists of all analytic functions  on D such that         B  := sup ∈D (1 − || 2 )         ()      < ∞. (2) Refer to [3] for more details on -Bloch spaces.
It is easy to check that ‖‖  := |(0)| + ‖‖ B  is a norm on B  , and B  is a Banach space equipped with this norm (see, e.g., [4]).Clearly, B  includes B  as its special case.Indeed, if () = ( For  ∈ D, let   () = ( − )/(1 − ) be the involutive automorphism of the unit disc which interchanges  and 0. We recall that in [5], for  ≥ 0,  ∈  (D)   ()) Q  is a Banach space, and Q ,0 is the closure of all polynomials in Q  .It is well known that Q 1 = BMOA, the space of all analytic functions of the bounded mean oscillation on D. Q 0 is the classical Dirichlet space D. For all 1 <  < ∞, Q  is the Bloch space B. Also, Q 1,0 = VMOA, the subspace of BMOA consisting of all analytic functions with vanishing mean oscillation, and for  > 1, Q ,0 = B 0 ; see [5,6] for more details on those spaces.
Let  1 and  2 be two linear subspaces of (D).If  is an analytic self-map of D, then  induces a composition operator   :  1 →  2 defined by Composition operators have been studied by numerous authors in many subspaces of (D).Among others, Madigan and Matheson characterized the continuity and compactness of composition operators on the classical Bloch space B in [7].Lou studied composition operators on Q  spaces in [8].
Composition operators between the logarithmic Bloch-type space and Q  log are studied in [9][10][11].This paper studies composition operators from -Bloch type spaces B  to Q  spaces.After some necessary background materials, Section 2 gives some function-theoretic characterizations of bounded and compact composition operators   : B  → Q  by using the Hadamard gap series technique.Section 3 characterizes the continuity, compactness of   : B  → Q ,0 , and the Fredholmness of   on Q ,0 .
Throughout the paper we use the same letter  to denote various positive constants which may change at each occurrence.Variables indicating the dependency of constants  will be often specified in the parenthesis.We use the notation  ≲  or  ≳  for nonnegative quantities  and  to mean  ≤  for some inessential constant  > 0. Similarly, we use the notation  ≈  if both  ≲  and  ≲  hold.

Composition Operators from B 𝜇 to Q 𝑝
We recall that an analytic function  on the unit disk D has Hadamard gaps if with  +1 /  ≥  > 1 for all  ∈ N. The following results are cited from [12].
In the sequel, we always suppose that  is as in Theorem B. The next lemma will play a key role in our main results.
On the other hand, in the disc || ≤ 1 −  −1 , we have that   (0) = 0,   (0) ̸ = 0, and   and   have a finite number of zeros in the disc.Hence if   and   have common zeros in the disc || ≤ 1 −  −1 , then one can replace  by the function  0 () = (  ) for an appropriate  and obtain a pair of functions which satisfy inequality (13).
We now characterize the boundedness of the composition operator   : Theorem 2. Let  > 0 and  be an analytic self-map of the unit disc.Then   : Proof.Assume that   : Now, we are going to characterize the compactness of composition operators   : B  → Q  .In [13], Tjani showed the following result.Lemma 3. Let ,  be two Banach spaces of analytic functions on D. Suppose the following: (1) The point evaluation functions on  are continuous.
(2) The closed unit ball of  is compact subset of  in the topology of uniform convergence on compact sets. ( Since   → 0 uniformly on the compact subsets of the unit disc, as  → ∞, then    → 0 as  → ∞ uniformly on the compact subsets of the unit disc.So for every  > 0, there is  0 ∈ N such that, for each  >  0 ,  1 ≤ , and By (29) there exists  0 ∈ (0, 1) such that, for every  >  0 ,  3 ≤ .Thus ‖  (  )‖ Q  → 0 as  → ∞, which completes the proof by Lemma 4.
The following corollary is an immediate result of Theorems 2 and 5.
The following theorem characterizes the equivalence of boundedness and compactness of composition operators from B  to Q ,0 .Theorem 8. Let 0 <  < ∞ and  is an analytic self-map of D. Then the following are equivalent.
(1)   : Finally, we consider the Fredholmness of composition operators on Q ,0 spaces.For a Banach space , recall that a bounded linear operator  on  is said to be Fredholm if both the dimension of its kernel and the codimension of its image are finite.This occurs if and only if  is invertible modulo the compact operators; that is, there is a bounded linear operator  such that both − and − are compact.We also notice that an operator is Fredholm if and only if its dual is Fredholm (see, e.g., [15]).
Before giving our result on Fredholmness, we need a useful result due to Wirths and Xiao [16].
(3)  belongs to the closure of the class of the polynomials in the norm ‖ ⋅ ‖ Q  .
Theorem 10.Let  be an analytic self-map of the unit disc D.
Then the following are equivalent.
(1)  is a Möbius transformation of D.
Assume  is not one to one.So there exist  1 , Recently, many authors have studied different classes of Bloch type spaces, where the typical weight function (1−|| 2 )  is replaced by a continuous positive function  defined on D.More precisely, let  : D → (0, ∞) be a radial weight function; that is, () = (||),  ∈ D, which is decreasing in a neighborhood of 1, continuous and lim || → 1 − (||) = 0.The Bloch type space B  consists of all  ∈ (D) such that         B  := sup ∈D  ()        ()      < ∞.

Lemma 1 .
There exist two functions ,  ∈ B  such that )  :  →  is continuous when  and  are given the topology of uniform convergence on compact sets.Then,  is a compact operator if and only if, given a bounded sequence {  } in  such that   → 0 uniformly on compact sets, the sequence {  } converges to zero in the norm of .We now use Lemma 4 above to give a characterization of compact composition operator   : B  → Q  .Let  > 0 and  be an analytic self-map of the unit disc.Then   : B  → Q  is compact if and only if  ∈ Q  and We first assume that   : B  → Q  is compact; then  ∈ Q  .Since ‖  /‖ B  ≲ 1 and   / → 0 as  → ∞, uniformly on any compact subsets of the unit disk, then by Lemma 4, ‖  (  /)‖ Q  → 0 as  → ∞.So for each  ∈ (0, 1) and each  > 0, there exists  0 ∈ N such that  2( 0 −1) sup Since ‖  ‖ B  ≲ 1 and   uniformly to  on any compact subsets of the unit disk, for  > 0 there exists  0 ∈ (0, 1) such that, for all  ≥  0 , as in the proof of Theorem 2. Conversely, we assume that  ∈ Q  and (29) holds.Let {  } ∈N be a sequence of functions in the unit ball of B  , such that   → 0 uniformly on the compact subsets of the unit disc as  → ∞.We notice that, for  ∈ (0, 1), so the point evaluation functionals on Q  are continuous.Thus, as a consequence of Lemma 3, we have the following result.Lemma 4. The composition operator  : B  → Q  is compact ifand only if for every bounded sequence {  } ∈N ⊆ B  , which converges uniformly to zero on any compact subset of the unit disk, ‖  (  )‖ Q  → 0 as  → ∞. 2  () = 0. (29) Proof.∈D ∫ {|()|>}        ()      2 (1 −       ()     2 )   () < 2.(31) We now consider the functions   () = () and  ∈ (0, 1) for  with ‖‖ B  ≤ 1. ∈D ∫ {|()|>}        ()      2 (1 −       ()     2 ) Namely, for each ‖‖ B  ≤ 1 and  > 0, there is 0 <  < 1 and some constant () depending only on  such that, for  ∈ [, 1),   () <  () .(34) Since   is compact, it maps the unit ball of B  into a relative compact subset of Q  .Thus for each  > 0, there exists a finite collection of functions  1 ,  2 , . . .,   in the unit ball of B  , such that for each ‖‖ B  ≤ 1 there is a  ∈ {1, 2, . . ., } with sup ∈D ∫ D      ( ∘ )  () − (  ∘ ) =:  1 +  2 +  3 .