# Essential Norms of Stević–Sharma Operators from General Banach Spaces into Zygmund-Type Spaces

Authors: M. A. Bakhit
Journal: Journal of Mathematics
 (2022)

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## Abstract

A Stević–Sharma operator denoted byTψ1,ψ2,φ is a generalization product of multiplication, differentiation, and composition operators. Using several restrictive terms, we characterize an approximation of the essential norm of the Stević–Sharma operator Tψ1,ψ2,φ from a general class X of holomorphic function spaces into Zygmund-type spaces with some of the most convenient test functions on the open unit disk. As an application, we show that our results hold up for several other domain spaces of Tψ1,ψ2,φ, such as the Hardy space and the weighted Bergman space.


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## Body

## 1. Introduction

LetD=ζ∈ℂ:ζ<1 be the open unit disk of the complex plane ℂ. We denote by HOD the family of analytic functions in D; also, we denote by ASD the family of analytic self-maps of D, and AutD the set of conformal automorphisms of the disk D. As usual, the Banach space H∞ is the family of bounded functions h∈HOD, defined by the norm h∞=supζ∈Dhζ<∞. For any weighted function μ, the weighted Banach space Hμ∞ is the space of functions h∈HOD such that(1)hμ∞=supζ∈Dhζμζ<∞.Moreover, the little weighted Banach spaceHμ,0∞ is the space of functions h∈HOD such that(2)lim|ζ|⟶1−hζμζ=0.The Bloch-type spaceℬμ is the Banach space of functions h∈HOD such that h′∈Hμ∞, with the norm hℬμ=h0+h′μ∞. The little Bloch-type space ℬμ,0 is the closed subspace of ℬμ consisting of functions h∈HOD such that h′∈Hμ,0∞.The space of all functionsh∈HOD is said to be a weighted Zygmund-type space Zμ if all functions h are such that h″∈Hμ,0∞. The space Zμ becomes a Banach space with the norm(3)hZμ=h0+h′0+supζ∈Dh″ζμζ.The little weighted Zygmund-type spaceZμ,0 is the closed subspace of Zμ and consists of functions h∈HOD, such that lim|ζ|⟶1−h″ζμζ=0.Exactly as in the aforementioned spaces, whenμζ=1−ζ2d, we get the Zygmund-type space Zd. In addition, the Zd space is a Banach space with the norm h0+h′0+hZd, where hZd=supζ∈Dh″ζ1−ζ2d<∞. The Zygmund spaces also satisfy Zd⊊Zl whenever 0<d<l. When d=1, we get the classical Zygmund space denoted by Z. In fact, the motivation for the name and for studying the Zygmund-type spaces comes from the Zygmund class; see, for example, Chapter 5 of Duren’s book [1]. The Zygmund space and its subspace Z0 play an important role in connection to the theory of the Hardy spaces Hq,0<q<1. Indeed, in [2], it was shown that Z can be viewed as the dual of the Hardy space Hq,q=1/2. More generally, the spaces defined by replacing the function h in the definition of Z with its n th derivative, under the assumption h,h′,…,hn∈HOD∩CD for n≥1, can be viewed as duals of the Hardy spaces Hq,q=1/n+2. A recent study of several operators on Zygmund-type spaces has attracted significant research attention; see, for example, [3–10].For anyΨ∈HOD and φ∈ASD, we have the following linear operators:Dhζ=h′ζ, the differentiation operatorMΨhζ=Ψζhζ, the multiplication operatorCφhζ=h°φζ=hφζ, the composition operatorWΨ,φhζ=Ψζ⋅hφζ, the weighted composition operatorFor a unified manner of treatment of these operators, many researchers seek to present various products of multiplication, composition, and differentiation operators; see, for example, [8, 11–18]. Stević et al. was the first to introduce the operator Tψ1,ψ2,φ in [19]. So, for any ψ1,ψ2∈HOD and φ∈ASD, the operator Tψ1,ψ2,φ is called the Stević–Sharma operator, which is defined as(4)Tψ1,ψ2,φhζ=ψ1ζhφζ+ψ2ζh′φζ,h∈HOD.Over the past 10 years, the boundedness and compactness of this operator have been studied extensively in the most well-known spaces of holomorphic functions; for example, see [3, 19–27].This paper proceeds as follows. In Section2, we present basic facts and prerequisites required for the general class X of holomorphic function spaces. In Section 3, we characterize upper and lower bounds for the essential norm of the Tψ1,ψ2,φ operators from X into the Zygmund-type space Zμ (or Zμ,0). Finally, in Section 4, we show that our estimations withstand any choice space, provided that it is reflexive, such as the Hardy spaces Hq,1≤q≤∞, and the weighted Bergman space Aβq (β>−1 and 1≤q<∞). This work is a continuation of our not yet published article regarding the boundedness and compactness of Tψ1,ψ2,φ between the currently considered spaces; see [20].In this paper, for any two quantitiesQ1h and Q2h that are jointly dependent on h∈HOD, we stipulate that Q1h≲Q2h, meaning that there is a positive constant C that fulfills Q1h≤CQ2h. Thus, when Q2h≲Q1h≲Q2h, we hold that Q1h≈Q2h and the quantities Q1h and Q2h are said to be equivalent. If Q1h≈Q2h, then Q2h<∞ if and only if Q1h<∞.


## 2. Preliminaries

Following popular terminology in functional analysis, we stipulate a Banach spaceX (or Y) whose elements are functions h∈HOD with the norm hX and whose the functionals of point-evaluation are bounded.LetEζ indicate the functional of point-evaluation at ζ. So, for any ζ∈D, we can define the norm(5)ΛXζ=suphζ:h∈X,hX≤1=EζX.For eachh∈X and ζ∈D, we obtain(6)hζ≤hXΛXζ.The following sets of conditions on Banach spaceX, given in [28], include the conditions used to formulate the results of the present work. For all s∈0,1 and ζ∈D, if h∈X, we obtainA1ΛXsζ≲ΛXζ, for all s∈0,1.A2 By a positive constant δ, the norm ΛXζ is bounded below in compact subsets of D and lim|ζ|⟶1ΛXζ=∞.A3 With respect to the uniform convergence topology on compact subsets of D, the unit ball BX of a Banach space X is comparatively compact.A4 For all h∈X and S∈AutD, we have Skh∈X and SkhX≲hX, where k=1,2,3.A5 For s∈0,1, let hsζ=hsζ,ζ∈D, where h∈X. Then, the linear map Cs:h⟶hs is a compact map on X.A6 For s∈0,1, sup0<s<1CsX<∞.A7 For all ζ∈D, let h∈X, then 1−ζ2khkζ≲hXΛXζ, where k=1,2,3.If a Banach spaceX contains polynomials and satisfies the conditions A3, A4, and A7, then X is said to be an admissible space (see [28–30]). If the set of polynomials in an admissible space X is dense, it is said to be polynomial dense. Examples of a polynomial dense space include the Hardy spaces Hq,1≤q≤∞ and the weighted Bergman space Aβq for all β>−1 and 1≤q<∞ (see [31]).The following proposition is Proposition1 in [28].Proposition 1.
Forζ∈D, the map ζ⟶ΛXζ is bounded on compact subsets of D in a Banach space X. Further, if a Banach space X is reflexive, then BX is compact and shows relatively of uniform convergence with respect to the topology on a compact subset of D.The following lemma helps to distinguish the properties of the operatorTψ1,ψ2,φ.Lemma 1.
Let there be a Banach spaceX satisfying the above conditions A4 and A7. Suppose that φ∈ASD, then for any w∈D and h∈X, such that hX≤1, there is a set of functions Fw,k defined in X, for k=1,2,3, such that supw∈DFw,kX<∞,(7)Fw,kjφw=0,ifk>j,and(8)Fw,kjφw=k!−1khφw1−φw2j,ifk=j.Moreover, the sequencesFw,k converge to 0 uniformly on compact subsets of D, when w⟶1.Proof.
Letρbζ=b−ζ/1−b¯ζ, which is the automorphism of D that changes zero and b, that is, ρb∈AutD, for any ζ∈D. Note that(9)ρb′b=−11−b2,ρb″b=−2b¯1−b22,and(10)ρb3b=4b¯21−b23.
Now, fixh∈X, such that hX≤1 and consider the functions Fw,kw=ρφwwkhw, for all k∈1,2,3 and w∈D. Let S=ρφw in condition A4, then for all k=1,2,3, we have Fw,k∈X and(11)Fw,kX=SkhX≲hX≲1.
Also, it is not difficult to prove that(12)Fw,1φw=0;Fw,1′φw=−hφw1−|φw|2;Fw,2φw=Fw,2′φw=0;Fw,2″φw=2hφw1−|φw|22;Fw,3φw=Fw,3′φw=Fw,3″φw;=0;Fw,33φw=−6hφw1−|φw|23.
Finally, the uniform convergence to 0 of the sequencesFw,k is self-evident when |w|⟶1.This work is a continuation of [20], where we describe the boundedness and the compactness of Tψ1,ψ2,φ:X⟶Zμ. The following boundedness result has been proven.(14)K0˜μ,ψ,φ=supw∈Dμw|ψ1″w|ΛXφw,K1˜μ,ψ,φ=supw∈Dμw2ψ1′wφ′w+ψ1wφ″w+ψ2″w1−|φw|2ΛXφw,K2˜μ,ψ,φ=supw∈Dμwψ1wφ′2w+2ψ2′wφ′w+ψ2wφ″w1−|φw|22ΛXφw,K3˜μ,ψ,φ=supw∈Dμwψ2wφ′2w1−|φw|23ΛXφw.Theorem 1.
Let there be a Banach spaceX satisfying the above conditions A4 and A7. Then, Tψ1,ψ2,φ:X⟶Zμ is a bounded operator if and only if all the quantities Kj˜μ,ψ,φ are finite, for j=0,1,2,3. Moreover, in which case(13)Tψ1,ψ2,φX⟶Zμ≈∑j=03Kj˜μ,ψ,φ,where the quantities Kj˜μ,ψ,φ,j=0,1,2,3 are defined as follows:The following lemma is proved in a standard way; see, for example [11, 28].Lemma 2.
Suppose thatX,Y are two Banach spaces satisfying the following conditions:B1 The point evaluation functionals Eζ are continuous on XB2 With respect to the uniform convergence topology on compact subsets of D, the closed unit ball B¯X of a Banach space X is comparatively compactB3 The operator T is continuous from X into Y when X and Y are given the topology of uniform convergence on compact setsThen, the bounded operatorT:X⟶Y is compact if and only if for any bounded sequence hm in X converges uniformly to 0 as m⟶∞ on compact subsets of D, and we have limn⟶∞‖Thm‖Y=0.We know that the essential norm‖T‖e of an operator T is its distance from the compact operators in the operator norm. Specifically, let X and Y be two Banach spaces and let T:X⟶Y be a bounded linear operator, then the essential norm of T between X and Y is referred to as ‖T‖e,X⟶Y, and its definition is(15)Te,X⟶Y=infT−TX⟶Y|T:X⟶Yis compact.Lemma 3 (see [28]).
LetX be a Banach space of functions h∈HOD, such that ‖h‖X≤1. Fix any ε>0,r∈0,1 and choose s∈0,1.(a)
If the Banach spaceX includes some functions nonvanishing at zero and fulfill either A2 or A6, then(16)sup‖h‖X≤1supζ∈DI−CshζΛXζ<∞.(b)
If the Banach spaceX satisfies one of the following sets of conditions:(i)
BothA1 and A5(ii)
A5 and X is reflexive(iii)
A7then(17)sup‖h‖X≤1sup|ζ|≤rI−Cshζ<ε.(c)
If the Banach spaceX satisfies A2 and one of the conditions in part (b), then(18)sup‖h‖X≤1supζ∈DI−CshkζΛXζ<ε,∀k=1,2.Remark 1.
Fix anyε>0,r∈0,1 and choose s∈0,1. By part (c) of Lemma 3, we have I−Csh3 converging uniformly to 0 on compact subsets as s⟶1−. Since X satisfies A2, under the same conditions, we obtain(19)sup‖h‖X≤1supζ∈DI−Csh3ζΛXζ<ε,Remark 2.
For anyφ∈ASD, if we suppose that(20)infw∈DΛXφw>0,we observe that assumption (20) is sufficient to apply Lemma 3, while part (c) of Lemma 3 requires condition A2.


## 3. Essential Norm ofTψ1,ψ2,φ:X⟶Zμ

Here, we provide an approximation of the essential norm ofTψ1,ψ2,φ from a large class X of Banach spaces into the Zygmund-type space Zμ, under specific conditions on class X on the open unit disk. With quantities Kjμ,ψ,φj=0,1,2,3 defined below, the following result complements Theorem 3 in [32]. To simplify the formulation of the main results in this section, for ΛXw as in (5) and δ>0, we set(21)K0μ,ψ,φ=limδ⟶1sup|φw|>δμw|ψ1″w|ΛXφw,K1μ,ψ,φ=limδ⟶1sup|φw|>δμw2ψ1′wφ′w+ψ1wφ″w+ψ2″w1−|φw|2ΛXφw,K2μ,ψ,φ=limδ⟶1sup|φw|>δμwψ1wφ′2w+2ψ2′wφ′w+ψ2wφ″w1−|φw|22ΛXφw,K3μ,ψ,φ=limδ⟶1sup|φw|>δμwψ2wφ′2w1−|φw|23ΛXφw.Theorem 2.
Let a Banach spaceX of functions in HOD be reflexive and satisfy the above conditions A4, A5, and A7, together with either A1 or A6. Suppose that infw∈DΛXφw>0, if Tψ1,ψ2,φ:X⟶Zμ is a bounded operator. Then,(22)Tψ1,ψ2,φe,X⟶Zμ≈∑j=03Kjμ,ψ,φ.Proof.
First, we prove that,(23)‖Tψ1,ψ2,φ‖e,X⟶Zμ≥rsim∑j=03Kjμ,ψ,φ.
If we suppose that‖φ‖∞<1, then it follows immediately that Kjμ,ψ,φ=0 for all j=0,1,2,3. This is why, hereafter, we suppose ‖φ‖∞=1, then we prove that Kjμ,ψ,φ≲Tψ1,ψ2,φe,X⟶Zμ for all j=0,1,2,3.
For eachm∈ℕ and fix M>0, we choose hm∈X such that ‖hm‖X≤1,(24)hmφbm>ΛXφbm−M.
By assuming thatX is reflexive, the unit ball BX of a Banach space X is compact under the uniform convergence topology on compact subsets of D. Using the fact that |hmw|≤‖hm‖XΛXw, then hm is a uniformly bounded sequence on compact sets of D. For all w∈D and m∈ℕ, set the functions(25)fmw=1−|φbm|2w1−φbm¯whmw.
Then, observe that(26)|fmw|=1−|φbm|21−|w|ΛXw,and(27)fmφbm=φbmhmφbm.
Clearlyfm converges uniformly to 0 on compact subsets of D. By the condition A4, we obtain fm∈X and ‖fm‖X≲‖hm‖X≲1.
Sincefm∈X, by the functions defined in Lemma 1, we consider the functions(28)Fm,kbm=ρφbmbmkfmbm,fork=1,2,3.
By Lemma1, observe that(29)Fm,kjφbm=0,ifk>jandFm,kjφbm=k!−1kΛXφbm1−|φbm|2j,ifk=j.
Further,Fm,k∈X and ‖Fm,k‖X≲‖fm‖X≲1, for all k=1,2,3.
Using the hypothesis that conditionsA4 and A7 hold, then all sequences Fw,k are bounded in a Banach space X. Moreover, the sequences Fw,k converge to 0 uniformly on compact subsets of D, when |w|⟶1.Step 1.
In the case ofj=3, let bm be a sequence in D, such that limm⟶∞φbm=1 and(30)K3μ,ψ,φ≔limm⟶∞μbmψ2bmφ′2bm1−|φbm|23ΛXφbm.
Letk=3 in (29), then we have(31)Fm,3φbm=Fbm,3′φbm=Fm,3″φbm=0,Fm,33φbm=−6fmφbm1−|φbm|23.
Thus,(32)Tψ1,ψ2,φFm,3″bm=ψ2bmφ′2bmFm,33φbm=6ψ2bmφ′2bmfmφbm1−|φbm|23.
Now, supposing thatT:X⟶Zμ is a compact operator, then the operator Tψ1,ψ2,φ−T is bounded. Therefore, by Proposition 1 and Lemma 2, we obtain(33)limm⟶∞TFm,3Zμ=0.
By (32) and (33), we have(34)‖Tψ1,ψ2,φ−T‖X⟶Zμ>˜limsupm⟶∞‖Fm,3‖X‖Tψ1,ψ2,φ−T‖X>˜limsupm⟶∞‖Tψ1,ψ2,φ−TFm,3‖Zμ=limsupm⟶∞‖Tψ1,ψ2,φFm,3‖Zμ≥limsupm⟶∞μbmTψ1,ψ2,φFm,3″bm=limsupm⟶∞μbm6ψ2bmφ′2bmfmφbm1−|φbm|23.
Using (24) and (27), we obtain(35)μbm6ψ2bmφ′2bm1−|φbm|23ΛXφbm−M≤μbm6ψ2bmφ′2bm1−|φbm|23hmφbm≤μbm6ψ2bmφ′2bm1−|φbm|23fmφbm.
Thus, by (30), (34), and (35), we have(36)K3μ,ψ,φ≔limm⟶∞μbmψ2bmφ′2bm1−|φbm|23ΛXφbm≲‖Tψ1,ψ2,φ−T‖X⟶Zμ≲infT‖Tψ1,ψ2,φ−T‖X⟶Zμ≲‖Tψ1,ψ2,φ‖e,X⟶Zμ.Step 2.
In the case ofj=2, we let a sequence bm in D be such that limm⟶∞|φbm|=1 and(37)K2μ,ψ,φ:=limm⟶∞μbmψ1bmφ′2bm+2ψ2′bmφ′bm+ψ2bmφ″bm1−|φbm|22ΛXφbm.
Lettingk=2 in (29), we observe that(38)Fm,2φbm=Fm,2′φbm=0,Fm,2″φbm=2fmφbm1−|φbm|22.
So, we have(39)=ψ1bmφ′2bm+2ψ2′bmφ′bm+ψ2bmφ″bmFm,2″φbm+ψ2bmφ′2bmFm,23φbm≥ψ1bmφ′2bm+2ψ2′bmφ′bm+ψ2bmφ″bmFm,2″φbm−ψ2bmφ′2bmFm,23φbm.
By Proposition1 and Lemma 2, if T:X⟶Zμ is a compact operator, then we obtain(40)limm⟶∞‖TFm,2‖Zμ=0.
By (39) and (40), we have(41)‖Tψ1,ψ2,φ−T‖X⟶Zμ>˜limsupm⟶∞‖Fm,2‖X‖Tψ1,ψ2,φ−T‖X>˜limsupm⟶∞‖Tψ1,ψ2,φ−TFm,2‖Zμ>˜limsupm⟶∞μbm|ψ1bmφ′2bm+2ψ2′bmφ′bm+ψ2bmφ″bm2fmφbm1−|φbm|22|−limsupm⟶∞μbm6ψ2bmφ′2bmfmφbm1−|φbm|23.
Hence,(42)limsupm⟶∞μbmψ1bmφ′2bm+2ψ2′bmφ′bm+ψ2bmφ″bm2fmφbm1−|φbm|22≲‖Tψ1,ψ2,φ−T‖X⟶Zμ+limsupm⟶∞μbm6ψ2bmφ′2bmfmφbm1−|φbm|23.
Using (24) and (36), then (42) gives(43)limsupm⟶∞μbmψ1bmφ′2bm+2ψ2′bmφ′bm+ψ2bmφ″bm1−|φbm|22ΛXφbm≲‖Tψ1,ψ2,φ−T‖X⟶Zμ+M+K3μ,ψ,φ.
SinceM is an arbitrary constant, combining (43) with (36) and (37), we conclude that(44)K2μ,ψ,φ≲‖Tψ1,ψ2,φ−T‖X⟶Zμ+K3μ,ψ,φ≲infT‖Tψ1,ψ2,φ−T‖X⟶Zμ+K3μ,ψ,φ≲‖Tψ1,ψ2,φ‖e,X⟶Zμ.Step 3.
In the case ofj=1, we let a sequence bm in D be such that limm⟶∞|φbm|=1 and(45)K1μ,ψ,φ≔limm⟶∞μbm2ψ1′bmφ′bm+ψ1bmφ″bm+ψ2″bm1−|φbm|2ΛXφbm.
Lettingk=1 in (29), we observe that(46)Fm,1φbm=0,Fm,1′φbm=−fmφbm1−|φbm|2.
Thus, we have(47)Tψ1,ψ2,φFm,1″bm=|2ψ1′bmφ′bm+ψ1bmφ″bm+ψ2″bmFm,1′φbm+ψ1bmφ′2bm+2ψ2′bmφ′bm+ψ2bmφ″bmFm,1″φbm+ψ2bmφ′2bmFm,13φbm|.
By Proposition1 and Lemma 2, if T:X⟶Zμ is a compact operator, then we obtain(48)limm⟶∞‖TFm,1‖Zμ=0.
By (47) and (50), we have(49)‖Tψ1,ψ2,φ−T‖X⟶Zμ>˜limsupm⟶∞‖Fm,1‖X‖Tψ1,ψ2,φ−T‖X>˜limsupm⟶∞‖Tψ1,ψ2,φ−TFm,1‖Zμ>˜limsupm⟶∞μbm2ψ1′bmφ′bm+ψ1bmφ″bm+ψ2″bm−fmφbm1−|φbm|2−limsupm⟶∞μbmψ1bmφ′2bm+2ψ2′bmφ′bm+ψ2bmφ″bm2fmφbm1−|φbm|22−limsupm⟶∞μbm6ψ2bmφ′2bmfmφbm1−|φbm|23.
Hence,(50)limsupm⟶∞μbm2ψ1′bmφ′bm+ψ1bmφ″bm+ψ2″bm−fmφbm1−|φbm|2≲‖Tψ1,ψ2,φ−T‖X⟶Zμ+limsupm⟶∞μbm6ψ2bmφ′2bmfmφbm1−|φbm|23+limsupm⟶∞μbmψ1bmφ′2bm+2ψ2′bmφ′bm+ψ2bmφ″bm2fmφbm1−|φbm|22.
Using (24), (36), (44), and then (50) gives(51)limsupm⟶∞μbm2ψ1′bmφ′bm+ψ1bmφ″bm+ψ2″bm1−|φbm|2ΛXφbm≲‖Tψ1,ψ2,φ−T‖X⟶Zμ+M+K3μ,ψ,φ+K2μ,ψ,φ.
SinceM is an arbitrary constant, combining (51) with (44) and (45), we conclude that(52)K1μ,ψ,φ≲‖Tψ1,ψ2,φ−T‖X⟶Zμ+K2μ,ψ,φ+K3μ,ψ,φ≲infT‖Tψ1,ψ2,φ−T‖X⟶Zμ+K2μ,ψ,φ+K3μ,ψ,φ≲‖Tψ1,ψ2,φ‖e,X⟶Zμ.Step 4.
In the case ofj=0, for each m∈ℕ and fix M>0, let hm and fm be such that fmφbm=φbmhmφbm in terms of the sequence bm in D such that limm⟶∞|φbm|=1 and(53)K0μ,ψ,φ:=limm⟶∞μbmψ1″bmΛXφbm.
Then,(54)Tψ1,ψ2,φfm″bm=ψ1″bmfmbm+|2ψ1′bmφ′bm+ψ1bmφ″bm+ψ2″bmfm′φbm+ψ1bmφ′2bm+2ψ2′bmφ′bm+ψ2bmφ″bmfm″φbm+ψ2bmφ′2bmfm3φbm|.
IfT:X⟶Zμ is a compact operator, then by Lemma 2, we obtain(55)limm⟶∞‖Tfm‖Zμ=0.
By (54) and (55), we have(56)‖Tψ1,ψ2,φ−T‖X⟶Zμ>˜limsupm⟶∞‖fm‖X‖Tψ1,ψ2,φ−T‖X>˜limsupm⟶∞‖Tψ1,ψ2,φ−Tfm‖Zμ>˜limsupm⟶∞μbmψ1″bmfmbm−limsupm⟶∞μbm2ψ1′bmφ′bm+ψ1bmφ″bm+ψ2″bm−fmφbm1−|φbm|2−limsupm⟶∞μbmψ1bmφ′2bm+2ψ2′bmφ′bm+ψ2bmφ″bm2fmφbm1−|φbm|22−limsupm⟶∞μbm6ψ2bmφ′2bmfmφbm1−|φbm|23.
Hence, by conditionA7 and (56), we have(57)limsupm⟶∞μbmψ1″bmfmφbm≲‖Tψ1,ψ2,φ−T‖X⟶Zμ+limsupm⟶∞μbm6ψ2bmφ′2bmfmφbm1−|φbm|23+limsupm⟶∞μbmψ1bmφ′2bm+2ψ2′bmφ′bm+ψ2bmφ″bm2fmφbm1−|φbm|22+limsupm⟶∞μbm2ψ1′bmφ′bm+ψ1bmφ″bm+ψ2″bm−fmφbm1−|φbm|2.
Using (6), (14), (21), and (50), then (57) gives(58)limsupm⟶∞μbmψ1″bmΛXφbm≲‖Tψ1,ψ2,φ−T‖X⟶Zμ+M+K3μ,ψ,φ+K2μ,ψ,φ+K1μ,ψ,φ.
SinceM is an arbitrary constant, combining (58) with (52) and (53), we conclude that(59)K0μ,ψ,φ≲‖Tψ1,ψ2,φ−T‖X⟶Zμ+K1μ,ψ,φ+K2μ,ψ,φ+K3μ,ψ,φ≲infT‖Tψ1,ψ2,φ−T‖X⟶Zμ+K1μ,ψ,φ+K2μ,ψ,φ+K3μ,ψ,φ≲‖Tψ1,ψ2,φ‖e,X⟶Zμ.
We thus obtain the lower estimate on the essential norm.
Now, let us prove the upper estimate. By the following,(60)‖Tψ1,ψ2,φ‖e,X⟶Zμ≲∑j=03Kjμ,ψ,φ.
In the statement of the theorem, we assume thatinfw∈DΛXφw>0, which is the same as assumption (20) in Remark 2. By reading carefully the detailed proof of Lemma 3 in [28], while part (c) of the lemma requires condition A2, we can conclude that the assumption infw∈DΛXφw>0 is sufficient to apply Lemma 3. Thus, we fix ε>0,r∈0,1 and choose s∈0,1 as in Lemma 3 and Remark 2.
By conditionA5, since Tψ1,ψ2,φ:X⟶Zμ is a bounded operator, the linear map Cs is compact on X, and the product Tψ1,ψ2,φCs:X⟶Zμ is compact. Therefore,(61)‖Tψ1,ψ2,φ‖e,X⟶Zμ≤‖Tψ1,ψ2,φ−Tψ1,ψ2,φCs‖e,X⟶Zμ=sup‖h‖X≤1‖Tψ1,ψ2,φI−Csh‖Zμ≤sup‖h‖X≤1Tψ1,ψ2,φI−Csh0+Tψ1,ψ2,φI−Csh′0+sup‖h‖X≤1∑j=03Ljμ,ψ,φ,where(62)L0μ,ψ,φ=supw∈Dμwψ1″wI−Cshφw,L1μ,ψ,φ=supw∈Dμw2ψ1′wφ′w+ψ1wφ″w+ψ2″wI−Csh′φw,L2μ,ψ,φ=supw∈Dμwψ1wφ′2w+2ψ2′wφ′w+ψ2wφ″wI−Csh″φw,L3μ,ψ,φ=supw∈Dμwψ2wφ′2wI−Csh3φw.
By Lemma3, we obtain(63)sup‖h‖X≤1Tψ1,ψ2,φI−Csh0+Tψ1,ψ2,φI−Csh′0≤sup‖h‖X≤1ψ10I−Cshφ0+ψ20I−Csh′φ0+sup‖h‖X≤1ψ1′0I−Cshφ0+ψ10φ′0I−Csh′φ0+sup‖h‖X≤1ψ2′I−Csh′φ0+ψ20φ′0I−Csh″φ0≤εψ10|+|ψ1′0+εψ20|+|ψ10φ′0+ψ2′0|+|ψ20φ′0ΛXφ0.
Now, combining conditionA7 with Lemma 3, Remark 1, Theorem 1, and (20), we obtain(64)sup‖h‖X≤1L0μ,ψ,φ≤sup‖h‖X≤1sup|φw|≤rμwψ1″wΛXφwI−CshφwΛXφw+sup‖h‖X≤1sup|φw|>rμwψ1″wΛXφwI−CshφwΛXφw≲εK0˜μ,ψ,φ+sup|φw|>rμwψ1″wΛXφw,(65)sup‖h‖X≤1L1μ,ψ,φ≤sup‖h‖X≤1sup|φw|≤rμw2ψ1′wφ′w+ψ1wφ″w+ψ2″w1−|φw|2·ΛXφw×1−|φw|2I−Csh′φwΛXφw+sup‖h‖X≤1sup|φw|>rμw2ψ1′wφ′w+ψ1wφ″w+ψ2″w1−|φw|2·ΛXφw×1−|φw|2I−Csh′φwΛXφw≲εK1˜μ,ψ,φ+sup|φw|>rμw2ψ1′wφ′w+ψ1wφ″w+ψ2″w1−|φw|2ΛXφw,(66)sup‖h‖X≤1L2μ,ψ,φ≤sup‖h‖X≤1sup|φw|≤rμwψ1wφ′2w+2ψ2′wφ′w+ψ2wφ″w1−|φw|22·ΛXφw×1−|φw|22I−Csh″φwΛXφw+sup‖h‖X≤1sup|φw|>rμwψ1wφ′2w+2ψ2′wφ′w+ψ2wφ″w1−|φw|22·ΛXφw×1−|φw|22I−Csh″φwΛXφw≲εK2˜μ,ψ,φ+sup|φw|>rμwψ1wφ′2w+2ψ2′wφ′w+ψ2wφ″w1−|φw|22ΛXφw,(67)sup‖h‖X≤1L3μ,ψ,φ≤sup‖h‖X≤1sup|φw|≤rμwψ2wφ′2w1−|φw|23ΛXφw×1−|φw|23I−Csh3φwΛXφw+sup‖h‖X≤1sup|φw|>rμwψ2wφ′2w1−|φw|23ΛXφw×1−|φw|23I−Csh3φwΛXφw≲εK3˜μ,ψ,φ+sup|φw|>rμwψ2wφ′2w1−|φw|23ΛXφw.
Using (38), (39), (40), and (67), we obtain(68)‖Tψ1,ψ2,φ‖e,X⟶Zμ≲ε+sup|φw|>rμwψ1″wΛXφw+sup|φw|>rμw2ψ1′wφ′w+ψ1wφ″w+ψ2″w1−|φw|2ΛXφw+sup|φw|>rμwψ1wφ′2w+2ψ2′wφ′w+ψ2wφ″w1−|φw|22ΛXφw+sup|φw|>rμwψ2wφ′2w1−|φw|23ΛXφw.
Sinceε is arbitrary, if ‖φ‖∞<1, then we have ‖Tψ1,ψ2,φ‖e,X⟶Zμ=0, and Tψ1,ψ2,φ:X⟶Zμ is compact. If ‖φ‖∞=1, letting r⟶1, we have(69)‖Tψ1,ψ2,φ‖e,X⟶Zμ≲∑j=03Kjμ,ψ,φ.Remark 3.
Note that the assumption that the Banach spaceX is reflexive is used only in the case of a lower estimate of Theorem 2. This brings about an interesting question. What is the possibility of approximating the essential norm in Theorem 2 without assuming the reflectivity on the Banach space X?
Actually, by conditionB3 in Lemma 2, if Tψ1,ψ2,φ:X⟶Zμ is a compact operator and the sequence hm is bounded in the Banach space X converging to zero uniformly on compact subsets of D, then limn⟶∞‖Thm‖Zμ=0. Thus, our argument in the proof of Theorem 2 for hm taken to be Fm,k,k=1,2,3 leads to a lower estimate on ‖Tψ1,ψ2,φ‖e,X⟶Zμ. An example of a space satisfying condition B3 in Lemma 2 is the Hardy spaces H∞.
Thus, the approximation of‖Tψ1,ψ2,φ‖e,X⟶Zμ also holds if the conditions B1—B3 in Lemma 2 hold.In the casesψ1w=ψw and ψ2w=0 in Theorem 2, we deduce the following result as a corollary; see Theorem 3.3 in [4].(71)R0=limδ⟶1sup|φw|>δμw|ψ″w|ΛXφw,R1=limδ⟶1sup|φw|>δμw2ψ′wφ′w+ψwφ″w1−|φw|2ΛXφw,R2=limδ⟶1sup|φw|>δμwψwφ′2w1−|φw|22ΛXφw.Corollary 1.
Let a Banach spaceX of functions in HOD be reflexive and satisfy the above conditions A4, A5, and A7, together with either A1 or A6. Suppose that infw∈DΛXφw>0, if Tψ,φ:X⟶Zμ is a bounded operator. Then,(70)‖Tψ,φ‖e,X⟶Zμ≈∑j=02Rjμ,ψ,φ,where


## 4. Applications of Essential Norm on Other Domains

We now show that our estimations hold up for any domain space choice, provided that the domain is reflexive. Note that, all spaces handled below contain the constant functions. Thus, condition (20) applies to all cases. First, set the following quantities:(72)Q0μ,ψ,φ;α=μwψ″w1−|φw|2α,Q1μ,ψ,φ;α=μw2ψ1′wφ′w+ψ1wφ″w+ψ2″w1−|φw|2α+1,Q2μ,ψ,φ;α=μwψ1wφ′2w+2ψ2′wφ′w+ψ2wφ″w1−|φw|2α+2,Q3μ,ψ,φ;α=μwψ2wφ′2w1−|φw|2α+3.Second, we discuss the approximation of‖Tψ1,ψ2,φ‖e on other domain spaces.


### 4.1. The Hardy SpacesHq,1≤q≤∞

Observe that the mapζ⟶Λζ is a constant 1 mapping. Clearly for the H∞ space, all conditions A2–A5 hold. By Theorem 5.4 in [33], for any integer k, we know that the space H∞ and the Bloch space ℬ are connected by the condition as follows:(73)1−|ζ|k|hkζ|≲‖h‖ℬ,forζ∈D,h∈ℬ.Thus, theH∞ space also satisfies the condition A7. So, we deduce that the approximation of the essential norm ‖Tψ1,ψ2,φ‖e,H∞,X⟶Zμ is applicable. The theory below summarizes this approach, and the results are applicable for the domain space H∞.Theorem 3.
If the operatorTψ1,ψ2,φ:H∞⟶Zμ is a bounded operator, then(74)‖Tψ1,ψ2,φ‖e,H∞⟶Zμ≈∑j=03Qjμ,ψ,φ;0.Forq≥1 and h∈HOD, the Hardy space HqD=Hq is the Banach space with the norm as follows:(75)‖h‖Hqq=sup0<s<1∫02π|hseiθ|qdθ2π.We know that the explicit formula for Hardy spaceHq is given by(76)ΛHqζ=11−|ζ|21/q.Moreover, for a positive integerk,ζ∈D and 1≤q<∞ if h∈Hq, then(77)hkζ≲hHq1−|ζ|2k+1/q.From (76) and (77), we have that the conditions A4 and A7 hold for all k=1,2,3 on Hardy space Hq. Thus, the norm estimate of the operator is an outcome instant from Theorem 1.The results forTψ1,ψ2,φ:Hq⟶Zμ,0 are an instant outcome from Theorem 2. We can then summarize the results as follows.Theorem 4.
IfTψ1,ψ2,φ:Hq⟶Zμ is a bounded operator, then(78)Tψ1,ψ2,φe,Hq⟶Zμ≈∑j=03Qjμ,ψ,φ;α,where α=1/q.


### 4.2. The Weighted Bergman SpaceAβq

For allβ>−1 and 1<q<∞, the weighted Bergman space Aβq is the set of all functions h∈HOD, such that(79)‖h‖Aβqq=∫D|hζ|qdAβζ<∞,where dAβζ=β+11−|ζ|2βdAζ is the weighted Lebesgue measure. When β>−1, the positive measure dAβ is normalized to become a probability measure. The weighted Bergman-type space Aβq is a Banach space when 1≤q<∞, and also it is a complete metric space when 0<q<1. Several properties of Aβq spaces are discussed in [31, 34].The following lemma is proved in [31] and Lemma 5.3 in [4].Lemma 4.
For a positive integerk and ζ∈D, let β>−1 and 1≤q<∞ and let h∈Aβq. Then,(i)
(80)ΛAβqζ=sup|hζ|:h∈Aβq,‖h‖Aβq≤1=11−|ζ|2β+2/q.(ii)
|hkζ|≲‖h‖Aβq/1−|ζ|2k+β+2/q.Let us summarize‖Tψ1,ψ2,φ‖e from the domain space Aβq.Theorem 5.
IfTψ1,ψ2,φ:Aβq⟶Zμ is a bounded operator, then(81)‖Tψ1,ψ2,φ‖e,Aβq⟶Zμ≈∑j=03Qjμ,ψ,φ;α,where α=β+2/q.Proof.
From Lemma4, we have that conditions A4 and A7 hold. Thus, the norm estimate of the operator is an outcome instant from Theorem 1.
For the essential norm estimates of‖Tψ1,ψ2,φ‖Aβq⟶Zμ, we observe that conditions A1–A3 and A5 hold for Aβq,1<q<∞ (see [28]). Since Aβq is a reflexive Banach space if 1<q<∞, in the case of an arbitrary weight μ, the hypotheses of Theorem 2 hold.


## 4.1. The Hardy SpacesHq,1≤q≤∞

Observe that the mapζ⟶Λζ is a constant 1 mapping. Clearly for the H∞ space, all conditions A2–A5 hold. By Theorem 5.4 in [33], for any integer k, we know that the space H∞ and the Bloch space ℬ are connected by the condition as follows:(73)1−|ζ|k|hkζ|≲‖h‖ℬ,forζ∈D,h∈ℬ.Thus, theH∞ space also satisfies the condition A7. So, we deduce that the approximation of the essential norm ‖Tψ1,ψ2,φ‖e,H∞,X⟶Zμ is applicable. The theory below summarizes this approach, and the results are applicable for the domain space H∞.Theorem 3.
If the operatorTψ1,ψ2,φ:H∞⟶Zμ is a bounded operator, then(74)‖Tψ1,ψ2,φ‖e,H∞⟶Zμ≈∑j=03Qjμ,ψ,φ;0.Forq≥1 and h∈HOD, the Hardy space HqD=Hq is the Banach space with the norm as follows:(75)‖h‖Hqq=sup0<s<1∫02π|hseiθ|qdθ2π.We know that the explicit formula for Hardy spaceHq is given by(76)ΛHqζ=11−|ζ|21/q.Moreover, for a positive integerk,ζ∈D and 1≤q<∞ if h∈Hq, then(77)hkζ≲hHq1−|ζ|2k+1/q.From (76) and (77), we have that the conditions A4 and A7 hold for all k=1,2,3 on Hardy space Hq. Thus, the norm estimate of the operator is an outcome instant from Theorem 1.The results forTψ1,ψ2,φ:Hq⟶Zμ,0 are an instant outcome from Theorem 2. We can then summarize the results as follows.Theorem 4.
IfTψ1,ψ2,φ:Hq⟶Zμ is a bounded operator, then(78)Tψ1,ψ2,φe,Hq⟶Zμ≈∑j=03Qjμ,ψ,φ;α,where α=1/q.


## 4.2. The Weighted Bergman SpaceAβq

For allβ>−1 and 1<q<∞, the weighted Bergman space Aβq is the set of all functions h∈HOD, such that(79)‖h‖Aβqq=∫D|hζ|qdAβζ<∞,where dAβζ=β+11−|ζ|2βdAζ is the weighted Lebesgue measure. When β>−1, the positive measure dAβ is normalized to become a probability measure. The weighted Bergman-type space Aβq is a Banach space when 1≤q<∞, and also it is a complete metric space when 0<q<1. Several properties of Aβq spaces are discussed in [31, 34].The following lemma is proved in [31] and Lemma 5.3 in [4].Lemma 4.
For a positive integerk and ζ∈D, let β>−1 and 1≤q<∞ and let h∈Aβq. Then,(i)
(80)ΛAβqζ=sup|hζ|:h∈Aβq,‖h‖Aβq≤1=11−|ζ|2β+2/q.(ii)
|hkζ|≲‖h‖Aβq/1−|ζ|2k+β+2/q.Let us summarize‖Tψ1,ψ2,φ‖e from the domain space Aβq.Theorem 5.
IfTψ1,ψ2,φ:Aβq⟶Zμ is a bounded operator, then(81)‖Tψ1,ψ2,φ‖e,Aβq⟶Zμ≈∑j=03Qjμ,ψ,φ;α,where α=β+2/q.Proof.
From Lemma4, we have that conditions A4 and A7 hold. Thus, the norm estimate of the operator is an outcome instant from Theorem 1.
For the essential norm estimates of‖Tψ1,ψ2,φ‖Aβq⟶Zμ, we observe that conditions A1–A3 and A5 hold for Aβq,1<q<∞ (see [28]). Since Aβq is a reflexive Banach space if 1<q<∞, in the case of an arbitrary weight μ, the hypotheses of Theorem 2 hold.



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