# A New Characterization of Generalized Weighted Composition Operators from the Bloch Space into the Zygmund Space

Authors: Hao Li; Xiaohong Fu
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

A new criterion for the boundedness and the compactness of the generalized weighted composition operators from the Bloch space into the Zygmund space is given in this paper.


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## Body

## 1. Introduction

LetH(𝔻) be the space of analytic functions on the unit disk 𝔻={z:|z|<1}. An f∈H(𝔻) is said to belong to the Bloch space, denoted by ℬ, if
(1)∥f∥ℬ=|f(0)|+supz∈𝔻(1-|z|2)|f′(z)|<∞.
Let ℬ0 be the little Bloch space, which consists of those f∈ℬ such that (1-|z|2)|f′(z)|→0, as |z|→1. Denote by 𝒵 the set of all f∈H(𝔻)⋂‍C(𝔻¯) such that
(2)∥f∥=sup|f(ei(θ+h))+f(ei(θ-h))-2f(eiθ)|h<∞,
where the supremum is taken over all eiθ∈∂𝔻 and h>0. In fact, f∈𝒵 if and only if supz∈𝔻(1-|z|2)|f′′(z)|<∞ by Theorem 5.3 of [1] and the Closed Graph Theorem. The class 𝒵 with the norm
(3)∥f∥𝒵=|f(0)|+|f′(0)|+supz∈𝔻(1-|z|2)|f′′(z)|
is a Banach space. We call 𝒵 the Zygmund space. For more information on the Zygmund space, see, for example, [1, 2].LetS(𝔻) denote the set of analytic self-map of 𝔻. Associated with φ∈S(𝔻) is the composition operator Cφ, which is defined by Cφ(f)(z)=f(φ(z)) for z∈𝔻 and f∈H(𝔻). We refer the readers to the book [3] for the theory of the composition operator on various function spaces.Letu∈H(𝔻). The weighted composition operator, denoted by uCφ, is defined by
(4)(uCφ)(f)(z)=u(z)f(φ(z)),z∈𝔻,f∈H(𝔻).Letn be a nonnegative integer. The generalized weighted composition operator, denoted by Dφ,un, is defined as follows (see [4–7]):
(5)(Dφ,unf)(z)=u(z)·f(n)(φ(z)),z∈𝔻,f∈H(𝔻).
When n=0, then Dφ,un becomes the weighted composition operator. If n=0 and u(z)=1, then Dφ,un=Cφ. If n=1 and u(z)=φ′(z), then Dφ,un=DCφ. If n=1 and u(z)=1, then Dφ,un=CφD. The operators DCφ and CφD were studied, for example, in [8–15].Composition operators, weighted composition operators, and related operators on the Bloch space were studied in [10, 12–22], while composition operators and weighted composition operators between the Zygmund space and some other spaces were studied in [2, 12, 23–28].Recently, many researchers studied the generalized weighted composition operatorDφ,un on various spaces; see, for example, in [4–7, 25, 26, 29–32]. In [26], Stević has studied the operator Dφ,un from the Bloch space to the nth weighted-type space, which includes the Zygmund space. Among other things, he obtained the following result (see [26]).Theorem A.
Letu∈H(𝔻), φ, an analytic self-map of  𝔻 and n a positive integer. Then the following assertions hold.(a)
The operatorDφ,un:ℬ(orℬ0)→𝒵 is bounded if and only if
(6)M1:=supz∈𝔻(1-|z|2)|u′′(z)|(1-|φ(z)|2)n<∞,M2:=supz∈𝔻(1-|z|2)|2u′(z)φ′(z)+u(z)φ′′(z)|(1-|φ(z)|2)n+1<∞,M3:=supz∈𝔻(1-|z|2)|u(z)||φ′(z)|2(1-|φ(z)|2)n+2<∞.(b)
Suppose that the operatorDφ,un:ℬ(orℬ0)→𝒵 is bounded then Dφ,un:ℬ(orℬ0)→𝒵 is compact if and only if
(7)lim|φ(z)|→1(1-|z|2)|u′′(z)|(1-|φ(z)|2)n=0,lim|φ(z)|→1(1-|z|2)|u(z)||φ′(z)|2(1-|φ(z)|2)n+2=0,lim|φ(z)|→1(1-|z|2)|2u′(z)φ′(z)+u(z)φ′′(z)|(1-|φ(z)|2)n+1=0.Here we give a new criterion for the boundedness or compactness of the operatorDφ,un; namely, we use three families of functions to characterize the operator Dφ,un:ℬ→𝒵.Throughout the paper,C denotes a positive constant which may differ from one occurrence to the other. The notation A≍B means that there exists a positive constant C such that B/C≤A≤CB.


## 2. Main Results

In this section we give our main results and proofs. Fora∈𝔻, set
(8)fa,j(z)=(1-|a|21-a¯z)j,j=1,2,3.
Next, we will use these three families of functions to characterize generalized weighted composition operators Dφ,un:ℬ→𝒵.Theorem 1.
Letu∈H(𝔻), φ be an analytic self-map of 𝔻, and n a positive integer. Then the following conditions are equivalent:(a)
the operatorDφ,un:ℬ→𝒵 is bounded;(b)
the operatorDφ,un:ℬ0→𝒵 is bounded;(c)
u∈𝒵, N2:=supz∈𝔻(1-|z|2)|u(z)||φ′(z)|2<∞, N3:=supz∈𝔻(1-|z|2)|2u′(z)φ′(z)+u(z)φ′′(z)|<∞,
(9)A≔max{supw∈𝔻∥Dφ,unfφ(w),1∥𝒵,supw∈𝔻∥Dφ,unfφ(w),2∥𝒵,supw∈𝔻∥Dφ,unfφ(w),3∥𝒵}<∞.Proof.
(a)⇒(b). This implication is obvious.
(b)⇒(c). Assume that Dφ,un:ℬ0→𝒵 is bounded. Taking the functions zn, zn+1, and zn+2 and using the boundedness of Dφ,un, we see that u∈𝒵, and N2 and N3 are finite. For each a∈𝔻, it is easy to check that fa,j∈ℬ0,j=1,2,3. Moreover ∥fa,j∥ℬ,j=1,2,3, are bounded by constants independent of a. By the boundedness of Dφ,un:ℬ0→𝒵, we get
(10)supa∈𝔻∥Dφ,unfφ(a),j∥𝒵≤∥Dφ,un∥supa∈𝔻∥fφ(a),j∥ℬ≤C∥Dφ,un∥<∞,j=1,2,3,
as desired.
(c)⇒(a). Suppose that u∈𝒵, and N2 and N3 and A are finite. To prove this implication, we only need to show that these conditions imply (6). A calculation shows that
(11)fa,1(n)(a)=n!a¯n(1-|a|2)n,fa,2(n)(a)=(n+1)!a¯n(1-|a|2)n,fa,3(n)(a)=(n+2)!a¯n2(1-|a|2)n.
For the simplicity, we denote 2u′(z)φ′(z)+u(z)φ′′(z) by v(z). From (11), for w∈𝔻, we have
(12)(Dφ,unfφ(w),1)′′(w)=n!u′′(w)φ(w)¯n(1-|φ(w)|2)n+(n+1)!v(w)φ(w)¯n+1(1-|φ(w)|2)1+n+(n+2)!u(w)(φ′(w))2φ(w)¯n+2(1-|φ(w)|2)2+n,(13)(Dφ,unfφ(w),2)′′(w)=(n+1)!u′′(w)φ(w)¯n(1-|φ(w)|2)n+(n+2)!v(w)φ(w)¯n+1(1-|φ(w)|2)1+n+(n+3)!u(w)(φ′(w))2φ(w)¯n+2(1-|φ(w)|2)2+n,(14)(Dφ,unfφ(w),3)′′(w)=(n+2)!u′′(w)φ(w)¯n2(1-|φ(w)|2)n+(n+3)!v(w)φ(w)¯n+12(1-|φ(w)|2)1+n+(n+4)!u(w)(φ′(w))2φ(w)¯n+22(1-|φ(w)|2)2+n.
Multiplying (12) by -(n+1) and then adding (13), we get
(15)-(n+1)(Dφ,unfφ(w),1)′′(w)+(Dφ,unfφ(w),2)′′(w)=(n+1)!v(w)φ(w)¯n+1(1-|φ(w)|2)1+n+2(n+2)!u(w)(φ′(w))2φ(w)¯n+2(1-|φ(w)|2)2+n.
Multiplying (12) by -(n+1)(n+2) and (14) by 2, respectively, we obtain
(16)-(n+1)(n+2)(Dφ,unfφ(w),1)′′(w)+2(Dφ,unfφ(w),3)′′(w)=2(n+2)!v(w)φ(w)¯n+1(1-|φ(w)|2)1+n+(4n+10)(n+2)!u(w)(φ′(w))2φ(w)¯n+2(1-|φ(w)|2)2+n.
Multiplying (15) by 2(n+2), we get
(17)-2(n+1)(n+2)(Dφ,unfφ(w),1)′′(w)+2(n+2)(Dφ,unfφ(w),2)′′(w)=2(n+2)!v(w)φ(w)¯n+1(1-|φ(w)|2)1+n+(4n+8)(n+2)!u(w)(φ′(w))2φ(w)¯n+2(1-|φ(w)|2)2+n.
Subtracting (17) from (16), we obtain
(18)2(n+2)!u(w)(φ′(w))2φ(w)¯n+2(1-|φ(w)|2)2+n=(n+1)(n+2)(Dφ,unfφ(w),1)′′(w)-2(n+2)(Dφ,unfφ(w),2)′′(w)+2(Dφ,unfφ(w),3)′′(w),
which implies that
(19)(1-|w|2)|u(w)(φ′(w))2||φ(w)|n+2(1-|φ(w)|2)2+n≤12n!(1-|w|2)|(Dφ,unfφ(w),1)′′(w)|+1(n+1)!(1-|w|2)|(Dφ,unfφ(w),2)′′(w)|+1(n+2)!(1-|w|2)|(Dφ,unfφ(w),3)′′(w)|(20)≤12n!∥Dφ,unfφ(w),1∥𝒵+1(n+1)!∥Dφ,unfφ(w),2∥𝒵+1(n+2)!∥Dφ,unfφ(w),3∥𝒵(21)≤A2n!+A(n+1)!+A(n+2)!.
From (16) and (18), we obtain
(22)(n+2)!v(w)φ(w)¯n+1(1-|φ(w)|2)1+n=-(n+1)(n+2)(n+3)(Dφ,unfφ(w),1)′′(w)+(n+2)(2n+5)(Dφ,unfφ(w),2)′′(w)-2(n+2)(Dφ,unfφ(w),3)′′(w),
which implies that
(23)(1-|w|2)|v(w)||φ(w)|n+1(1-|φ(w)|2)1+n≤(n+3)n!(1-|w|2)|(Dφ,unfφ(w),1)′′(w)|+(2n+5)(n+1)!(1-|w|2)|(Dφ,unfφ(w),2)′′(w)|+2(n+1)!(1-|w|2)|(Dφ,unfφ(w),3)′′(w)|(24)≤(n+3)n!∥Dφ,unfφ(w),1∥𝒵+(2n+5)(n+1)!∥Dφ,unfφ(w),2∥𝒵+2(n+1)!∥Dφ,unfφ(w),3∥𝒵(25)≤(n+3)n!A+(2n+5)(n+1)!A+2(n+1)!A.
By (12), (18), and (22), we have
(26)n!u′′(w)φ(w)¯n(1-|φ(w)|2)n=2+(n+1)(n+4)2(Dφ,unfφ(w),1)′′(w)-(n+3)(Dφ,unfφ(w),2)′′(w)+(Dφ,unfφ(w),3)′′(w),
which implies that
(27)(1-|w|2)|u′′(w)||φ(w)|n(1-|φ(w)|2)n≤2+(n+1)(n+4)2n!(1-|w|2)|(Dφ,unfφ(w),1)′′(w)|+(n+3)n!(1-|w|2)|(Dφ,unfφ(w),2)′′(w)|+1n!(1-|w|2)|(Dφ,unfφ(w),3)′′(w)|(28)≤2+(n+1)(n+4)2n!∥Dφ,unfφ(w),1∥𝒵+(n+3)n!∥Dφ,unfφ(w),2∥𝒵+1n!∥Dφ,unfφ(w),3∥𝒵(29)≤2+(n+1)(n+4)2n!A+(n+3)n!A+1n!A.
Fix r∈(0,1). If |φ(w)|>r, then from (21) we obtain
(30)(1-|w|2)|u(w)||φ′(w)|2(1-|φ(w)|2)2+n≤A2n!rn+2+A(n+1)!rn+2+A(n+2)!rn+2<∞.
On the other hand, if |φ(w)|≤r, we get
(31)(1-|w|2)|u(w)||φ′(w)|2(1-|φ(w)|2)2+n≤1(1-r2)2+nsupz∈𝔻(1-|z|2)|u(z)||φ′(z)|2<∞.
From (30) and (31) we see that M3 is finite. Similarly, from (25) and (29) we can obtain that M1 and M2 are finite as well. The proof of this theorem is finished.To get the characterization of the compactness ofDφ,un:ℬ(orℬ0)→𝒵, we need the following criterion, which follows from standard arguments similar to those outlined in Proposition 3.11 of [3].Lemma 2.
Letu∈H(𝔻), φ, an analytic self-map of 𝔻 and n be a positive integer. The operator Dφ,un:ℬ(orℬ0)→𝒵 is compact if and only if Dφ,un:ℬ(orℬ0)→𝒵 is bounded, and for any bounded sequence (fk)k∈ℕ in ℬ(orℬ0) which converges to zero uniformly on compact subsets of 𝔻, one has ∥Dφ,unfk∥𝒵→0 as k→∞.Theorem 3.
Letu∈H(𝔻), φ be an analytic self-map of 𝔻, and n a positive integer. Suppose that the operator Dφ,un:ℬ→𝒵 is bounded; then the following conditions are equivalent:(a)
the operatorDφ,un:ℬ→𝒵 is compact;(b)
the operatorDφ,un:ℬ0→𝒵 is compact;(c)
lim|φ(w)|→1∥Dφ,unfφ(w),j∥𝒵=0,j=1,2,3.Proof.
(a)⇒(b). This implication is clear.
(b)⇒(c). Assume that Dφ,un:ℬ0→𝒵 is compact. Let {wk}k∈ℕ be a sequence in 𝔻 such that limk→∞|φ(wk)|=1 (if such a sequence does not exist, then the limits in (c) automatically hold). Since the sequences {fφ(wk),j}, j=1,2,3, are bounded in ℬ0 and converge to 0 uniformly on compact subsets of 𝔻, by Lemma 2, we get ∥Dφ,unfφ(wk),j∥𝒵→0, as k→∞, which means that (c) holds.
(c)⇒(a). Suppose that the limits in (c) are 0. To prove this implication, we only need to show that (7) hold. Using the inequality (28), we get
(32)(1-|w|2)|u′′(w)|(1-|φ(w)|2)n≤[2+(n+1)(n+4)]∥Dφ,unfφ(w),1∥𝒵2n!|φ(w)|n+(n+3)∥Dφ,unfφ(w),2∥𝒵n!|φ(w)|n+∥Dφ,unfφ(w),3∥𝒵n!|φ(w)|n⟶0,
as |φ(w)|→1.
Using inequality (24), we get
(33)(1-|w|2)|2u′(w)φ′(w)+u(w)φ′′(w)|(1-|φ(w)|2)1+n≤(n+3)∥Dφ,unfφ(w),1∥𝒵n!|φ(w)|n+1+(2n+5)∥Dφ,unfφ(w),2∥𝒵(n+1)!|φ(w)|n+1+2∥Dφ,unfφ(w),3∥𝒵(n+1)!|φ(w)|n+1⟶0,
as |φ(w)|→1.
Using inequality (20), we get
(34)(1-|w|2)|u(w)(φ′(w))2|(1-|φ(w)|2)2+n≤∥Dφ,unfφ(w),1∥𝒵2n!|φ(w)|n+2+∥Dφ,unfφ(w),2∥𝒵(n+1)!|φ(w)|n+2+∥Dφ,unfφ(w),3∥𝒵(n+2)!|φ(w)|n+2⟶0,
as |φ(w)|→1. The desired result follows. The proof of this theorem is complete.



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