# Topological Structures of Derivative Weighted Composition Operators on the Bergman Space

Authors: Ce-Zhong Tong; Cheng Yuan; Ze-Hua Zhou
Journal: Journal of Function Spaces
 (2015)

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## Abstract

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.


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## Body

## 1. Introduction

LetD be the unit disk in the complex plane. The algebra of all holomorphic functions on domain D will be denoted by H(D). Let S(D) be the set of analytic self-maps of D. Every φ∈S(D) induces the composition operator Cφ defined by Cφf=f∘φ for f∈HD.Letu:D→C be analytic, and the weighted composition operator, uCφ, is defined by (1)uCφfz=uz·fφzfor any f∈H(D) and z∈D. When u≡1, it becomes a composition operator.Derivative weighted composition (d-composition) operator is defined by (2)Dφf=φ′f∘φ.It is obvious that Dφ=φ′Cφ.Forα>-1 and 0<p<∞, we recall that a weighted Bergman space Aαp(D) is the set of holomorphic functions on the unit disk for which (3)∫Dfzp1-z2αdAzπ<∞,where dA(z) is the Lebesgue area measure on the unit disk and fp is the pth root of this integral. Moreover, the Bergman space A2(D)=A02(D) is a Hilbert space with the inner product (4)f,g=∫Dfzgz¯dAzπ.A weighted Dirichlet spaceDαp is the set of holomorphic functions on the unit disk for which (5)∫Df′zp1-z2αdAzπ<∞with norm given by (6)fDαp=f0p+∫Df′zp1-z2αdAzπand the Dirichlet space D=D02 is also a Hilbert space with the inner product (7)f,g=f0g0¯+∫Df′zg′z¯dAzπ.It is easy to show that (Exercise 2.1.6 in [1]) Aα2(D)=Dα+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 [5–8].Another area of particular interest is the topological structure of the space of composition operators acting on a given function space. WhenX is a Banach space of analytic functions, we write C(X) for the space of composition operators and Cd(X) for the space of derivative weighted composition operators on X under the operator norm topology. The investigation of the topological structure of C(H2) was initiated by Berkson [9] in 1981. Central problem focuses on the relations between the structure of C(H2) and the compactness properties of its members.Continuing the work, in 1989, MacCluer [10] showed that, on the weighted Bergman space Aα2 for α≥-1, the compact composition operators form an arcwise connected set in C(Aα2) 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 C(H2)?In 2008, Gallardo-Gutiérrez et al. [12] gave a negative answer to the question for a variety of spaces in addition to Hp. In 2003, Bourdon [13] determined two linear-fractional self-maps of the disk having the same first-order data at a point ξ on the boundary of the disk and different second derivatives at ξ lie in the same component of C(H2), 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 Aα2, α>-1, and gave a partial answer to the component structure of C(Aα2). 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 CN.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.Theorem 1 (Theorem1 in [5]).
Letφ be an analytic self-map of D. Then the d-weighted composition operator Dφ is bounded on A2 if and only if(8)∫D1-z22φ′w21-z¯φw4dAw<∞.Building on those foundations, the present paper continues this line of research. The remainder is assembled as follows. In Section2, 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 Cd(A2), and some similar results about composition operators on the Dirichlet space are also presented there. In Section 3, we show that the d-composition operators form an arcwise connected set in Cd(A2). Finally, we determine when linear-fractional d-composition operators belong to the same component on the Bergman space under the operator norm topology in Section 4.


## 2. Difference of Derivative Weighted Composition Operators

Our discussion begins with the characterization of the compact difference ofd-composition operators. When we accomplished the following theorem, we read a new paper [22], in which Allen et al. obtained the same estimation (Lemma 10). Nonetheless, we present our version in the following, since our method is totally different.Theorem 2.
Supposeφ:D→D has finite angular derivative φ′(ζ) at some point ζ in the unit circle. Let ψ:D→D and consider the difference of Dφ and Dψ acting on the Bergman space A2(D). Then we have Dφ-Dψe2≥1 if the symbol ψ satisfies one of the following two conditions:(i)
ψ(ζ)≠φ(ζ) (as radial limits).(ii)
ψ(ζ)=φ(ζ) but ψ′(ζ)≠φ′(ζ).Proof.
Since the Bergman spaceA2(D) is rotationally invariant, we may assume ζ=1, φ(1)=1, and φ′(1)=s<∞. We first estimate Dφ-Dψ by considering the adjoint of Dφ-Dψ, (Dφ-Dψ)∗ acting on reproducing kernel functions Kz(w)=(1-z¯w)-2. Since(9)Dφ-Dψ∗Kz2=Dφ-Dψ∗Kz,Dφ-Dψ∗Kz=φ′z¯Kφz-ψ′z¯Kψz,φ′z¯Kφz-ψ′z¯Kψz=φ′z2Kφz2+ψ′z2Kψz2-2Reφ′z¯ψ′zKφzψz,the formulas for the kernel functions Kz and their norms show that(10)Dφ-Dψ2≥Dφ-Dψ∗Kz2Kz2=φ′z21-z21-φz22+ψ′z21-z21-ψz22-2Reφ′z¯ψ′z1-z21-φz¯ψz2for all z in D.
By the Julia-Carathéodory Theorem,(1-|z|2)/(1-|φ(z)|2) has nontangential limit 1/s as z goes to 1, so the first term of the right-hand side in the above expression has nontangential limit 1 as z goes to 1.
To deal with the third term of the right-hand side in (10) we distinguish two cases.
Case 1. Consider limr→1ψ(r)≠1. We can find zn approaching 1 nontangentially so that limn→∞ψ(zn)=u≠1.
From Schwarz-Pick Theorem we have that|ψ′(zn)|≤(1-|ψ(zn)|2)/(1-|zn|2) for every n=1,2,….
It follows that(11)2Reφ′zn¯ψ′zn1-zn21-φzn¯ψzn2≤2φ′znψ′zn1-zn21-φzn¯ψzn2≤2φ′zn1-φzn¯ψzn21-ψzn21-zn2has limit 0 as n tends to infinity since (12)limn→∞⁡φ′zn=s,limn→∞⁡1-φzn¯ψzn=1-u,limn→∞⁡1-zn2=0.
Combining with the above discussions and by (10), we can conclude that Dφ-Dψ2≥1.
Case 2. Consider ψ(1)=1 but ψ′(1)≠φ′(1)=s. We first consider(13)ψ′z-11-φz¯ψz1-z22=ψ′z-11-φz21-z2+φz¯φz-ψz1-z22=ψ′z-1/21-φz21-z2+φz¯1-z1-z2ψ′z-1/21-ψz1-z-ψ′z-1/21-φz1-z2.Let γM={z∈D:|1-z|/(1-|z|2)=M}, the boundary of a nontangential approach region at 1. As z approaches 1 along γM, the Julia-Carathéodory Theorem shows that (14)1-φz21-z2⟶s,1-φz1-z⟶sand either(15)1-ψz1-z⟶ψ′1whenψ′1<∞or(16)1-ψz1-z⟶∞whenψ′1=∞.
It is easy to show thatψ′(z)-1/21-ψz/1-z-ψ′(z)-1/2((1-φ(z))/(1-z))↛0. For any given positive number N, we may, by choosing M large enough, find a sequence zn approaching 1 along γM so that for n large enough (17)φzn¯1-zn1-zn2ψ′zn-1/21-ψzn1-zn-1-φzn1-zn>N+1.It follows from (13) that (18)ψ′zn-11-φzn¯ψzn1-zn22>N+1for n large enough. So, for arbitrary positive δ, choose N such that 2s/N<δ; we have (19)2Reφ′zn¯ψ′zn1-zn21-φzn¯ψzn2≤2sN<δfrom which and (10), we deduce that Dφ-Dψ2≥1-δ for arbitrary positive δ, so Dφ-Dψ2≥1 by letting δ→0.
It is standard to check the lower bound of the essential normDφ-Dψe2:(20)Dφ-Dψe2=inf⁡Dφ-Dψ-Q2:Qcompact≥limz→1⁡Dφ-Dψ∗KzKz≥1.This completes the proof of the theorem.As a direct consequence, we can obtain the following corollary.Corollary 3.
Ifφ has finite angular derivative on a set of positive measures, then Dφ is isolated in Cd(A2). For any ψ≠φ, Dφ-Dψe2≥1.Definition 4.
Letφ and ψ be two self-mappings of the unit disk, and we say that φ and ψ have thesame first-order boundary data or satisfy SBD, provided that whenever either |φ(ζ)|=1 or |ψ(ζ)|=1, then φ(ζ)=ψ(ζ) and φ′(ζ)=ψ′(ζ).Theorem 5.
SupposeDψ is in the component of Cd(A2) containing Dφ. Then φ and ψ must have the same first-order boundary data, where φ has finite angular derivative.Proof.
Suppose either|φ(ζ)|=1 or |ψ(ζ)|=1 at ζ in the unit circle. If ψ(ζ)≠φ(ζ) or ψ(ζ)=φ(ζ) but φ′(ζ)≠ψ′(ζ), by Theorem 2, Dφ-Dψe2≥1, Dψ is not in the component of Cd(A2) containing Dφ. Hence ψ(ζ)=φ(ζ) and φ′(ζ)≠ψ′(ζ); that is, φ and ψ must have the same first-order boundary data.d-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 C to denote a finite positive number which is not necessarily the same at each occurrence.Lemma 6.
Forα>-1 and 1<p≤q<∞, let φ,ψ∈Dαp be such that φ(D)⊂D and ψ(D)⊂D. Then Cφ-Cψ is bounded (compact) from Dαp into Dαq if and only if Dφ-Dψ is bounded (compact) from Aαp into Aαq.Proof.
We just verify the condition of boundedness. SupposeCφ-Cψ is bounded from Dαp into Dαq. Then there exists a constant C>0 such that (21)Cφ-CψfDαq≤CfDαpfor all f∈Dαp. Take f∈Aαp and let the function g∈Dαp be such that g′=f and g(0)=0. Then we have (22)Dφ-DψfAαqq=φ′f∘φ-ψ′f∘ψAαqq=g∘φ-g∘ψ′Aαqq=g∘φ-g∘ψDαqq=Cφ-CψgDαqq≤CgDαq=CfAαqq.Thus Dφ-Dψ:Aαp→Aαq is bounded.
Conversely, assumeDφ-Dψ:Aαp→Aαq is bounded. Take f∈Dαp such that f(0)=0. Then we have (23)Cφ-CψfDαq=f∘φ-f∘ψ′Aαqq=Dφ-Dψf′Aαqq≤Cf′Aαpp=CfDαp.Theorem 7.
Supposeφ is an analytic self-map of unit disk and has finite angular derivative φ′(ζ) at some point ζ in the unit circle. Let ψ be another analytic self-map of unit disk and consider Cφ and Cψ acting on D. Then unless both(1)
φ(ζ)=ψ(ζ),(2)
φ′(ζ)=ψ′(ζ),we have Cφ-Cψe2≥C, where the constant C depends only on the spaces D.Proof.
Note thatD=D02 and A2=A02. Setting α=0 and p=q=2 in Lemma 6, from the proof of Lemma 6, we know there exist two constants C1 and C2 such that (24)C1Dφ-DψA2≤Cφ-CψD≤C2Dφ-DψA2.The desired result now follows from Theorem 2.Corollary 8.
Ifφ has finite angular derivative on a set of positive measures, then Cφ is isolated in C(D). For any ψ≠φ, Cφ-Cψe2≥C, where constant C depends only on the D.Corollary 9.
SupposeCψ is in the component of C(D) containing Cφ. Then φ and ψ must have the same first-order boundary data, where φ has finite angular derivative.


## 3. Compact Composition Operators on Dirichlet Space

The proof of the following lemma is routine, which is omitted here.Lemma 10.
Letφ be an analytic self-map of D such that the induced composition operator Cφ is bounded on D. Then Cφ acts on D compactly if and only if whenever {fn} is a bounded sequence in D converging to zero uniformly on compact subsets of D, then Cφ(fn)→0.It is well known (e.g., Proposition  9.9 in [1]) that the compact composition operators on Dα form an arcwise connected set in C(Dα) for all α≥1. By Lemma 10, we find the result is also true on the Dirichlet D=D0 if we note that if φ∞<1, then Cφ is compact on the Dirichlet space.Theorem 11.
The compact composition operators onD 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.
GivenCφ and Cψ compact on D we first construct a continuous map of [0,1] into C(D) taking 1 to Cφ and 0 to Cφ0, the composition operator whose symbol is the constant map φ(0). The path t↦Cφt, where φt(z)=φ(tz), will satisfy our demand (φt∞<1 implying the compactness of Cφt).
Secondly, we construct continuous arc joining.
C
φ
0 to Cψ0. Let L(t)=(1-t)φ(0)+tψ(0) be the line segment in D from φ(0) to ψ(0) and then map [0,1] into C(D) by t↦CL(t), where CL(t)f(z)=f(L(t)). It is easy to verify that this map is continuous.It follows from Lemma6 and Theorem 11 that we have following corollary.Corollary 12.
The compactd-composition operators on A2(D) form an arcwise connected set in Cd(A2).From Theorem11, 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 H∞, while the answer is negative on classical Hardy space.Remark 13.
Letφ∈S(D) be a linear-fractional self-map of the unit disk with φ(ζ)=η, where both ζ and η are unimodular. Then the induced composition operator Cφ cannot be in the component generated by compact composition operators on Dirichlet space. Suppose Cφ did. The linear-fractional self-map φ must have finite angular derivative at ζ. And the composition operator induced by z/2 is compact on D. By Theorem 11 compact composition operators must be in the same component and Cφ and Cz/2 are in the same component. By Corollary 9  φ and z/2 must satisfy SBD. That is impossible.Lemma 14 (Corollary2 in [5]).
Letu be an analytic function on D and let φ be an analytic self-map of D. Then the weighted composition operator uCφ is compact on A2 if and only if lim|a|→1(uCφ)ka2=0, where ka(z) is the normalized reproducing kernel of A2 at a.There is a practical corollary of Lemma14.Lemma 15 (Proposition1 in [5]).
Letu be an analytic function on D and φ∈S(D). If the weighted composition operator uCφ is compact on A2, then (25)limz→1⁡1-z2uz1-φz2=0.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 inC(D) through the characterization of the component in Cd(A2).


## 4. Linear-Fractionald-Composition Operators

Throughout this section,φ and ψ will denote linear-fractional self-mappings of D. To determine when linear-fractional d-composition operators belong to the same component of Cd(A2), we need to state some lemmas.Lemma 16 (Theorem4.2.9 in [3]).
Supposen≥1 and f is analytic in D; then f∈A2(D) if and only if (1-|z|2)nf(n)(z) is in A2(D).Ifφ and ψ are linear-fractional self-maps of the unit disk with the same boundary data, φ(ζ)=ψ(ζ)=η, where ζ and η are in the unit circle. Then it is trivial that Dφ and Dψ are in the same path component in Cd(A2) if and only if Dη¯φ(ζz) and Dη¯ψ(ζz) are in the same path component in Cd(A2).So we can just discuss the situations whenφ(1)=1 and φ′(1)=s∈R. The following lemma follows Lemma  3.2 in [13] and its proof.Lemma 17.
Suppose thatφ is a nonautomorphic, linear-fractional self-mapping of D such that φ(1)=1 and φ′(1)=s. Denote T(z)=(1+z)/(1-z) for z∈D; then there is a complex constant a with positive real part such that (26)φz=T-11sTz+a=-1+as-sz-1-as+ss-1+asz-s-1-as.Conversely if we putφ(z)=((-1+as-s)z-1-as+s)/((s-1+as)z-s-1-as), it is clear that φ is a linear-fractional self-mapping of D such that φ(1)=1 and φ′(1)=s. By Lemma 15 and direct computation, it is easy to check all Cφ are noncompact on D. So all Dφ are also noncompact on A2 equivalently by Lemma 6.Lemma 18.
Letr:[0,1]→C be defined by r(t)=a+t(b-a) so that r is the straight-line path in the open right half-plane joining a and b. Let φt be the self-map of D given by(27)φtz=T-11sTz+rt=-1+rts-sz-1-rts+ss-1+rtsz-s-1-rts.Then t↦Dφt is continuous from the unit interval to Cd(A2), placing both Dφ and Dψ in the same path component.Proof.
Lett1 and t2 be distinct points in [0,1] with t1≤t2. This lemma will be proved if we proved that there is constant M such that (28)Dφt2-Dφt12≤Mt2-t12.We require some preliminary estimates. Fix z in the closure of D, let φz(t)=φt(z), and calculate the derivative of φz with respect to t: (29)φz′t=2b-as21-z2s-1+rtsz-s-1-rts2.For any z∈D¯ and t∈[0,1], note that |r(t)|≤|a|+|b|; we have(30)s-1+rtsz+s+1+rts≤2s+1+a+bs,s-1+rtsz-s-1-rts≥s+1+rts-s-1+rts.Since Re(r(t))>0, (31)s+1+rts2-s-1+rts2=4s+4Rerts>4s,it follows from (30) that(32)s-1+rtsz-s-1-rts≥s+1+rts2-s-1+rts2s-1+rtsz+s+1+rts≥4s2s+1+a+bs=2ss+1+a+bs.Combining (29) and (32), there is a constant K1=s|b-a|(s+1+s(|a|+|b|)) such that(33)φz′t≤K11-z2for all z∈D¯ and t∈[0,1]. Calculate the derivative of φt with respect to z: (34)φt′z=4ss-1+rtz-s-1-rts2.Similarly we have(35)φt′z≤K2for a constant K2 independent of z∈D¯ and t∈[0,1].
An easy calculation shows that(36)1-z1-φtz=s+1+rts-s-1+rtsz2s.Thus, applying the triangle inequality, we see that, for every z∈D, (37)1-z1-φtz≤s+1+a+bss=K3.
Becauseφt(D) is a proper subdisk of D internally tangent to the unit circle at 1, we may set (38)φtD=w∈D:w-w0<R,where 1-w0=R; that is, w0=1-R. It is easy to show that (39)w∈φtD⟺w-1-R2<R2⟺1-w2<R1-R1-w2,and this implies that(40)1-φtz21-φtz2<R1-Rfor any z∈D.
It follows from (27) that T(φt(z))=T(z)/s+r(t). And the image of the circle(41)1-w21-w2=R1-Runder T(z)=(1+z)/(1-z) is the line Re(r(t))+ki, where k∈R in right half-plane, and we put T-1(Re(r(t))+ki) back into (41). We have that (42)1-RR=1-T-1Rert+ki21-T-1Rert+ki2=4Rert+1+ki2-Rert-1+ki2=1Rert.Consequently,(43)1-φtz21-φtz2≤Rert-1≤min⁡Rea,Reb-1=K4.
For anyf∈A2, we have that (44)fφt2z-fφt1z=∫t1t2f′φztφz′tdt.This produces the crucial estimate: (45)Dφt2-Dφt1f2=∫Dφt2′zfφt2z-φt1′zfφt1z2dAz=∫Dφt2′z-φt1′zfφt2z+φt1′zfφt2z-fφt1z2dAz≤2∫D1-φt1′zφt2′z2φt2′z2fφt2z2dAz+2∫Dφt1′z2fφt2z-fφt1z2dAz=2I1+I2,I1=∫D1-s-1+rt2sz-s-1-rt2ss-1+rt1sz-s-1-rt1s22φt2′z2fφt2z2dAz≤K54t2-t12∫Dφt2′z2fφt2z2dAz≤K54t2-t12f2,where the constant K5 denotes upper bound of |(s-1+r(t)s)z-s-1-r(t)s|-1 multiplied by an absolute constant: (46)I2≤K2∫Dfφt2z-fφt1z2dAz=K2∫D∫t1t2f′φztφz′tdt2dAz≤K2t2-t1∫D∫t1t2f′φztφz′t2dtdAz≤4b-a2s4K2t2-t1∫t1t2∫Df′φtz2φt′z21-z4dAzdt≤K34K42K6t2-t1∫t1t2∫Df′φtz2φt′z21-φtz22dAzdt≤K34K42K6t2-t1∫t1t2dt∫Df′w21-w22dAw=K34K42K6t2-t121-w2f′w2≤K34K42K6t2-t12Cf2,where K6=4|b-a|2s4K2 and last inequality follows by using Lemma 16. Thus we have (47)Dφt2-Dφt1f2≤2K54+K34K42K6Cf2t2-t12,where Kj (j=3,4,5,6) and C depend only on a,b,s. This completes the proof of the lemma.Theorem 19.
Suppose thatφ and ψ are linear-fractional self-mappings of D. The following are equivalent:(a)
Dφ and Dψ lie in the same component of Cd(A2).(b)
φ and ψ satisfy SBD.(c)
Dφ and Dψ are joined by a continuous path in Cd(A2).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).
Ifφ is an automorphism, then, for any ζ in the unit circle, |φ(ζ)|=1. That φ and ψ satisfy SBD implies ψ(ζ)=φ(ζ) for any ζ in the unit circle. It turns out that ψ=φ by Maximum Principle, and the conclusion is obvious.
Suppose thatφ is not an automorphism; then φ(D) is a proper subdisk of D; there are exactly two possibilities:(i)
|φ(ζ)|<1 for any ζ∈∂D.(ii)
There exists a pointζ∈∂D such that |φ(ζ)|=1.
Suppose that(i) applies and φ and ψ satisfy SBD. Then ψ also must map the closure of D into D. In this situation, both Cφ and Cψ are compact operators on D. By Lemma 17, we have both Dφ and Dψ are compact on A2(D). Compact d-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).
We may suppose thatφ(1)=1. Our setting condition is that ψ share the same first-order boundary data with φ so ψ(1)=1 and we denote s=φ′(1)=ψ′(1). By Lemma 17, there are two constants a and b satisfying Re(a)>0 and Re(b)>0, such that (48)Tφz=1sTz+a,Tψz=1sTz+b.
Denoteφt described by Lemma 18; then t↦Dφt is continuous from the unit interval to Cd(A2), placing both Dφ and Dψ in the same path component by Lemma 18. This ends the proof of the theorem.Next corollary is an obvious consequence of Theorem19 and gives a partial description of component in C(D).Corollary 20.
Suppose thatφ and ψ are linear-fractional self-mappings of D. The following are equivalent:(a)
Cφ and Cψ lie in the same component of C(D).(b)
φ and ψ satisfy SBD.(c)
Cφ and Cψ are joined by a continuous path in C(D).



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