Topological Structures of Derivative Weighted Composition Operators on the Bergman Space

We characterize the difference of derivative weighted composition operators on the Bergman space in the unit disk and determine when linear-fractional derivative weighted composition operators belong to the same component of the space of derivative weighted composition operators on the Bergman space under the operator norm topology.


Introduction
Let D be the unit disk in the complex plane.The algebra of all holomorphic functions on domain D will be denoted by (D).Let (D) be the set of analytic self-maps of D. Every  ∈ (D) induces the composition operator   defined by    =  ∘  for  ∈ (D).
Let  : D → C be analytic, and the weighted composition operator,   , is defined by for any  ∈ (D) and  ∈ D. When  ≡ 1, it becomes a composition operator.Derivative weighted composition (-composition) operator is defined by   () =    ∘ . ( It is obvious that   =     . For  > −1 and 0 <  < ∞, we recall that a weighted Bergman space    (D) is the set of holomorphic functions on the unit disk for which where () is the Lebesgue area measure on the unit disk and ‖‖  is the th root of this integral.Moreover, the Bergman space  2 (D) =  2 0 (D) is a Hilbert space with the inner product A weighted Dirichlet space D   is the set of holomorphic functions on the unit disk for which It is easy to show that (Exercise 2.1.6 in [1])  2  (D) = D +2 .
Much effort has been expended on characterizing those analytic maps which induce bounded or compact composition operators.Readers interested in this topic can refer to the books [2] by Shapiro, [1] by Cowen and MacCluer, and [3,4] by Zhu, which are excellent sources for the development of the theory of composition operators and function spaces, and the recent papers [5][6][7][8].
Another area of particular interest is the topological structure of the space of composition operators acting on a given function space.When  is a Banach space of analytic functions, we write () for the space of composition operators and   () for the space of derivative weighted composition operators on  under the operator norm topology.The investigation of the topological structure of ( 2 ) was initiated by Berkson [9] in 1981.Central problem focuses on the relations between the structure of ( 2 ) and the compactness properties of its members.
Continuing the work, in 1989, MacCluer [10] showed that, on the weighted Bergman space  2  for  ≥ −1, the compact composition operators form an arcwise connected set in ( 2  ) and gave necessary conditions for two composition operators to have compact difference.At about the same time, Shapiro and Sundberg [11] gave further results on compact difference and isolation and, among other things, posed the following fundamental question and conjectured that it had a positive answer: ( * ) Do the compact composition operators form a connected component of the set ( 2 )?
In 2008, Gallardo-Gutiérrez et al. [12] gave a negative answer to the question for a variety of spaces in addition to   .In 2003, Bourdon [13] determined two linear-fractional self-maps of the disk having the same first-order data at a point  on the boundary of the disk and different second derivatives at  lie in the same component of ( 2 ), while the induced composition operators do not have compact difference.In 2005, Moorhouse [14] answers the question of compact difference for composition operators acting on  2  ,  > −1, and gave a partial answer to the component structure of ( 2  ).Later, Kriete and Moorhouse [15] extended their study to general linear combinations.Saukko [16,17] obtained a complete characterization of bounded and compact differences between standard weighted Bergman spaces.Recently, Choe et al. [18,19] extend Moorhouses characterization to the unit polydisk and unit ball in C  .Topological structure of composition operators, which are from some analytic function spaces into the Bloch type spaces or the space of bounded analytic functions, has been studied intensively during the past decades.Interested readers can refer to [20,21] and the references therein.
In 2004 Čučkovič and Zhao [5] characterized the bounded, compact, and Schatten class weighted composition operators on the Bergman space by generalized Berezin transforms.Derivative weighted composition operators are a special class of the weighted composition operators.The boundedness is characterized as follows.
Building on those foundations, the present paper continues this line of research.The remainder is assembled as follows.In Section 2, we characterize the compact difference of two derivative weighted composition operators on the Bergman space in the unit disk and discuss the isolation and component structure of the derivative weighted composition operator in C  ( 2 ), and some similar results about composition operators on the Dirichlet space are also presented there.In Section 3, we show that the -composition operators form an arcwise connected set in C  ( 2 ).Finally, we determine when linear-fractional -composition operators belong to the same component on the Bergman space under the operator norm topology in Section 4.

Difference of Derivative Weighted Composition Operators
Our discussion begins with the characterization of the compact difference of -composition operators.When we accomplished the following theorem, we read a new paper [22], in which for all  in D.
To deal with the third term of the right-hand side in (10) we distinguish two cases.
has limit 0 as  tends to infinity since lim Combining with the above discussions and by (10), we can conclude that Let , the boundary of a nontangential approach region at 1.As  approaches 1 along   , the Julia-Carathéodory Theorem shows that It is easy to show that   () −1/2 ((1 − ())/(1 − )) −   () −1/2 ((1 − ())/(1 − ))   0. For any given positive number , we may, by choosing  large enough, find a sequence   approaching 1 along   so that for  large enough It follows from (13) that for  large enough.So, for arbitrary positive , choose  such that 2/ < ; we have This completes the proof of the theorem.
As a direct consequence, we can obtain the following corollary.-composition operators on the weighted Bergman space are highly connected with composition operators on weighted Dirichlet space.In the following, we will use the symbol  to denote a finite positive number which is not necessarily the same at each occurrence .Lemma 6.For  > −1 and

Compact Composition Operators on Dirichlet Space
The proof of the following lemma is routine, which is omitted here.
Lemma 10.Let  be an analytic self-map of D such that the induced composition operator   is bounded on D. Then   acts on D compactly if and only if whenever {  } is a bounded sequence in D converging to zero uniformly on compact subsets of D, then ‖  (  )‖ → 0.
It is well known (e.g., Proposition 9.9 in [1]) that the compact composition operators on D  form an arcwise connected set in C(D  ) for all  ≥ 1.By Lemma 10, we find the result is also true on the Dirichlet D = D 0 if we note that if ‖‖ ∞ < 1, then   is compact on the Dirichlet space.

Theorem 11. The compact composition operators on D form an arcwise connected set in C(D).
The proof of this theorem is almost the same as the proof of Proposition 9.9 in [1], so we just give an outline here.
Proof.Given   and   compact on D we first construct a continuous map of [0, 1] into C(D) taking 1 to   and 0 to   0 , the composition operator whose symbol is the constant map (0).The path   →    , where   () = (), will satisfy our demand (‖  ‖ ∞ < 1 implying the compactness of    ).
It follows from Lemma 6 and Theorem 11 that we have following corollary.
Corollary 12.The compact -composition operators on  2 (D) form an arcwise connected set in C  ( 2 ).
From Theorem 11, we find that the compact composition operators on D are in the same component, but we do not know whether there is any noncompact composition operator belonging to the component generated by compact ones on D. The answers to those similar questions on different analytic function spaces appear somewhat different.One has positive answers when the spaces are classical Bergman space, Bloch space, and  ∞ , while the answer is negative on classical Hardy space.
Remark 13.Let  ∈ (D) be a linear-fractional self-map of the unit disk with () = , where both  and  are unimodular.Then the induced composition operator   cannot be in the component generated by compact composition operators on Dirichlet space.Suppose   did.The linear-fractional self-map  must have finite angular derivative at .And the composition operator induced by /2 is compact on D. By Theorem 11 compact composition operators must be in the same component and   and  /2 are in the same component.By Corollary 9  and /2 must satisfy SBD.That is impossible.Lemma 14 (Corollary 2 in [5]).Let  be an analytic function on D and let  be an analytic self-map of D. Then the weighted composition operator   is compact on  2 if and only if lim || → 1 ‖(  )  ‖ 2 = 0, where   () is the normalized reproducing kernel of  2 at .
There is a practical corollary of Lemma 14.

Lemma 15 (Proposition 1 in [5]). Let 𝑢 be an analytic function on D and 𝜑 ∈ 𝑆(D). If the weighted composition operator 𝑢𝐶
From this lemma, in next section we will see that some linear-fractional composition operators cannot be compact on Dirichlet space, and we will discuss that component in C(D) through the characterization of the component in C  ( 2 ).

Linear-Fractional 𝑑-Composition Operators
Throughout this section,  and  will denote linear-fractional self-mappings of D. To determine when linear-fractional composition operators belong to the same component of C  ( 2 ), we need to state some lemmas.
Lemma 16 (Theorem 4.2.9 in [3]).Suppose  ≥ 1 and  is analytic in D; then  ∈  2 (D) if and only if If  and  are linear-fractional self-maps of the unit disk with the same boundary data, () = () = , where  and  are in the unit circle.Then it is trivial that   and   are in the same path component in C  ( 2 ) if and only if  () and  () are in the same path component in C  ( 2 ).
Proof.Let  1 and  2 be distinct points in [0, 1] with  1 ≤  2 .This lemma will be proved if we proved that there is constant  such that We require some preliminary estimates.Fix  in the closure of D, let   () =   (), and calculate the derivative of   with respect to : Since Re(()) > 0, it follows from (30) that where 1 −  0 = ; that is,  0 = 1 − .It is easy to show that and this implies that     1 −   () where   ( = 3, 4, 5, 6) and  depend only on , , .This completes the proof of the lemma.Suppose that (i) applies and  and  satisfy SBD.Then  also must map the closure of D into D. In this situation, both   and   are compact operators on D. By Lemma 17, we have both   and   are compact on  2 (D).Compact -composition operators are shown to belong to the same arcwise connected subset in Corollary 12. Thus, to complete the proof of the theorem, we need to handle case (ii).

Theorem 19 .
Suppose that  and  are linear-fractional selfmappings of D. The following are equivalent: (a)   and   lie in the same component of C  ( 2 ).(b)  and  satisfy SBD.(c)   and   are joined by a continuous path in C  ( 2 ).Proof.Clearly (c) implies (a).That (a) implies (b) is proved in Theorem 5. Thus to complete the proof of Theorem 19, we establish (b) implies (c).If  is an automorphism, then, for any  in the unit circle, |()| = 1.That  and  satisfy SBD implies () = () for any  in the unit circle.It turns out that  =  by Maximum Principle, and the conclusion is obvious.Suppose that  is not an automorphism; then (D) is a proper subdisk of D; there are exactly two possibilities: (i) |()| < 1 for any  ∈ D.

Corollary 20 .
there are two constants  and  satisfying Re() > 0 and Re() > 0, such that  ( ()  described by Lemma 18; then   →    is continuous from the unit interval to C  ( 2 ), placing both   and   in the same path component by Lemma 18.This ends the proof of the theorem.Next corollary is an obvious consequence of Theorem 19 and gives a partial description of component in C(D).Suppose that  and  are linear-fractional selfmappings of D. The following are equivalent: (a)   and   lie in the same component of C(D).(b)  and  satisfy SBD.(c)   and   are joined by a continuous path in C(D).
Then the -weighted composition operator   is bounded on  2 if and only if We first estimate ‖  −   ‖ by considering the adjoint of   −   , (  −   ) * (10)om which and(10), we deduce that ‖  −   ‖ 2 ≥ 1 −  for arbitrary positive , so ‖  −   ‖ 2 ≥ 1 by letting  → 0.It is standard to check the lower bound of the essential norm ‖  −   ‖ 2  :        − and only if   −   is bounded (compact) from    into    .Thus   −   :    →    is bounded.Conversely, assume   −   :    →    is bounded.Take  ∈ D   such that (0) = 0. Then we have      (  −   )       Proof.Note that D = D 2 0 and  2 =  2 0 .Setting  = 0 and  =  = 2 in Lemma 6, from the proof of Lemma 6, we know there exist two constants  1 and  2 such that for all  ∈ D   .Take  ∈    and let the function  ∈ D   be such that   =  and (0) = 0. Then we have      (  −   )           =         ∘  −    ∘           =      ( ∘  −  ∘ )      Theorem 7. Suppose  is an analytic self-map of unit disk and has finite angular derivative   () at some point  in the unit circle.Let  be another analytic self-map of unit disk and consider   and   acting on D. Then unless both (1) () = (), (2)   () =   (), we have ‖  −   ‖ 2  ≥ , where the constant  depends only on the spaces D. 1        −        2 ≤        −       D ≤  2        −        2 .(24) The desired result now follows from Theorem 2. Corollary 8.If  has finite angular derivative on a set of positive measures, then   is isolated in C(D).For any  ̸ = , ‖  −   ‖ 2  ≥ , where constant  depends only on the D. Corollary 9. Suppose   is in the component of C(D) containing   .Then  and  must have the same first-order boundary data, where  has finite angular derivative.