# Norm and Essential Norm of an Integral-Type Operator from the Dirichlet Space to the Bloch-Type Space on the Unit Ball

Authors: Stevo Stević
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

Operator norm and essential norm of an integral-type operator,
recently introduced by this author, from the Dirichlet space to the Bloch-type
space on the unit ball inℂn are calculated here.


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## Body

## 1. Introduction

Let𝔹n=𝔹 be the open unit ball in ℂn, 𝔹1=𝔻 the open unit disk in ℂ, H(𝔹) the class of all holomorphic functions on 𝔹, and H∞(𝔹),  the space consisting of all f∈H(𝔹) such that ∥f∥∞=supz∈𝔹|f(z)|<∞.For anf∈H(𝔹) with the Taylor expansion f(z)=∑αaαzα, let (1.1)ℜf(z)=∑α|α|aαzα
be the radial derivative of f, where α=(α1,…,αn) is a multi-index, |α|=α1+⋯+αn and zα=z1α1⋯znαn. Let α!=α1!⋯αn!.The Dirichlet space𝒟2(𝔹)=𝒟2 contains all f(z)=∑αaαzα∈H(𝔹), such that (1.2)∥f∥𝒟22:=|f(0)|2+∑α|α|α!|α|!|aα|2<∞.
The quantity ∥f∥𝒟2 is a norm on 𝒟2 which for n=1 is equal to usual norm (1.3)∥f∥𝒟2(𝔻)=(|f(0)|2+∫𝔻|f′(z)|2‍dA(z))1/2,
where dA(z)=(1/π)rdrdθ is the normalized area measure on 𝔻.The inner product, between two functions(1.4)f(z)=∑αaαzα,g(z)=∑αbαzα,
on 𝒟2 is defined by (1.5)〈f,g〉:=f(0)g(0)¯+∑α|α|α!|α|!aαb¯α.Forα≠0, let (1.6)eα(z)=|α|!|α|α!zα,z∈𝔹,
and e0(z)≡1, then it is easy to see that the family {eα} is an orthonormal basis for 𝒟2,  and hence the reproducing kernel Kw(z) for 𝒟2 is given by ([1]) as follows:(1.7)Kw(z)=1+∑α≠0|α|!|α|α!zαw¯α=1+ln11-〈z,w〉,
where 〈z,w〉=∑j=1nzjw¯j is the inner product in ℂn.Clearly for eachf∈𝒟2 and w∈𝔹, the next reproducing formula holds:  (1.8)f(w)=〈f,Kw〉.
Note that for f=Kw from (1.8), we obtain  (1.9)Kw(w)=∥Kw∥𝒟22=lne1-|w|2.Also, by the Cauchy-Schwarz inequality and (1.9), we have that, for each f∈𝒟2 and w∈𝔹, (1.10)|f(w)|=|〈f,Kw〉|≤∥f∥𝒟2∥Kw∥𝒟2=∥f∥𝒟2(lne1-|w|2)1/2.
Note that inequality (1.10) is exact since it is attained for f=Kw.The weighted-type spaceHμ∞(𝔹)=Hμ∞ ([2, 3]) consists of all f∈H(𝔹) such that (1.11)∥f∥Hμ∞:=supz∈𝔹μ(z)|f(z)|<∞,
where μ is a positive continuous function on 𝔹 (weight).The Bloch-type spaceℬμ(𝔹)=ℬμ consists of all f∈H(𝔹) such that (1.12)∥f∥ℬμ:=|f(0)|+supz∈𝔹μ(z)|ℜf(z)|<∞,
where μ is a weight.Letg∈H(𝔻), g(0)=0, and φ be a holomorphic self-map of 𝔹, then the following integral-type operator: (1.13)Pφg(f)(z)=∫01f(φ(tz))g(tz)dtt,z∈𝔹,f∈H(𝔹)
has been recently introduced in [4] and considerably studied (see, e.g., [5–8] and the related references therein). For some related operators, see also [9–16] and the references therein.Usual problem in this research area is to provide function-theoretic characterizations for whenφ and g induce bounded or compact integral-type operator on spaces of holomorphic functions. Majority of papers only find asymptotics of operator norm of linear operators. Somewhat concrete but perhaps more interesting problem is to calculate operator norm of these operators between spaces of holomorphic functions on various domains. Some results on this problem can be found, for example, in [3, 17–26] (for related results see also [7, 27–32]). Having published paper [3], we started with systematic investigation of methods for calculating operator norms of concrete operators between spaces of holomorphic function.Here,  we calculate the operator norm as well as the essential norm of the operatorPφg:𝒟2→ℬμ, considerably extending our recent result in note [33].


## 2. Auxiliary Results

In this section, we quote several auxiliary results which are used in the proofs of the main results.Lemma 2.1 (see [4]).
Letg∈H(𝔹), g(0)=0, and φ be a holomorphic self-map of 𝔹, then
(2.1)ℜPφg(f)(z)=g(z)f(φ(z)).The next Schwartz-type lemma ([34]) can be proved in a standard way. Hence, we omit its proof.Lemma 2.2.
Assume thatg∈H(𝔹),  g(0)=0, μ is a weight, and φ is an analytic self-map of 𝔹, then Pφg:𝒟2→ℬμ is compact if and only if Pφg:𝒟2→ℬμ is bounded and for any bounded sequence (fk)k∈ℕ  in 𝒟2 converging to zero uniformly on compacts of 𝔹 as k→∞, one has
(2.2)limk→∞∥Pφg(fk)∥ℬμ=0.Lemma 2.3.
Assume thatg∈H(𝔹), g(0)=0, μ is a weight, φ is an analytic self-map of 𝔹 such that ∥φ∥∞<1, and the operator Pφg:𝒟2→ℬμ is bounded, then Pφg:𝒟2→ℬμ is compact.Proof.
First note that sincePφg:𝒟2→ℬμ is bounded and f0(z)≡1∈𝒟2 by Lemma 2.1,  it follows that ℜPφg(f0)=g∈Hμ∞. Now, assume that (fk)k∈ℕ is a bounded sequence in 𝒟2 converging to zero on compacts of 𝔹 as k→∞, then we have
(2.3)∥Pφg(fk)∥ℬμ≤∥g∥Hμ∞supw∈φ(𝔹)|fk(w)|→0,
as k→∞, since φ(𝔹) is contained in the ball |w|≤∥φ∥∞ which is a compact subset of 𝔹, according to the assumption ∥φ∥∞<1. Hence, by Lemma 2.2, the operator Pφg:𝒟2→ℬμ is compact.


## 3. Operator Norm ofPφg:𝒟2→ℬμ

In this section, we calculate the operator norm ofPφg:𝒟2→ℬμ.Theorem 3.1.
Assume thatg∈H(𝔹),  g(0)=0,  and φ is a holomorphic self-map of 𝔹, then
(3.1)∥Pφg∥𝒟2→ℬμ=supz∈𝔹μ(z)|g(z)|(lne1-|φ(z)|2)1/2=:L.Proof.
Using Lemma2.1, reproducing formula (1.8), the Cauchy-Schwarz inequality, and finally (1.9), we get that, for each f∈𝒟2 and w∈𝔹,
(3.2)μ(w)|ℜPφg(w)|=μ(w)|g(w)||f(φ(w))|=μ(w)|g(w)||〈f,Kφ(w)〉|≤μ(w)|g(w)|∥f∥𝒟2∥Kφ(w)∥𝒟2=∥f∥𝒟2μ(w)|g(w)|(lne1-|φ(w)|2)1/2.
Taking the supremum in (3.2) over w∈𝔹 as well as the supremum over the unit ball in 𝒟2  and using the fact Pφg(f)(0)=0, for each f∈H(𝔹), which follows from the assumption g(0)=0, we get
(3.3)∥Pφg∥𝒟2→ℬμ≤L.
Now assume that the operatorPφg:𝒟2→ℬμ is bounded. From (1.9), we obtain that, for each w∈𝔹,
(3.4)(lne1-|φ(w)|2)1/2∥Pφg∥𝒟2→ℬμ=∥Kφ(w)∥𝒟2∥Pφg∥𝒟2→ℬμ≥∥Pφg(Kφ(w))∥ℬμ=supz∈𝔹μ(z)|g(z)||Kφ(w)(φ(z))|≥μ(w)|g(w)||Kφ(w)(φ(w))|.
From (1.9) and (3.4), it follows that
(3.5)L≤∥Pφg∥𝒟2→ℬμ.
Hence, if Pφg:𝒟2→ℬμ is bounded, then from (3.3) and (3.5) we obtain (3.1).
In the casePφg:𝒟2→ℬμ is unbounded, the result follows from inequality (3.3).


## 4. Essential Norm ofPφg:𝒟2→ℬμ

LetX and Y be Banach spaces, and let  L:X→Y be a bounded linear operator. The essential norm of the operator L:X→Y, ∥L∥e,X→Y, is defined as follows: (4.1)∥L∥e,X→Y=inf{∥L+K∥X→Y:KiscompactfromXtoY},
where ∥·∥X→Y denote the operator norm.From this and since the set of all compact operators is a closed subset of the set of bounded operators, it follows thatL is compact if and only if (4.2)∥L∥e,X→Y=0.Here, we calculate the essential norm of the operatorPφg:𝒟2→ℬμ.Theorem 4.1.
Assume thatg∈H(𝔹),  g(0)=0,μ is a weight, φ is a holomorphic self-map of 𝔹, and Pφg:𝒟2→ℬμ is bounded. If ∥φ∥∞<1, then ∥Pφg∥e,𝒟2→ℬμ=0, and if ∥φ∥∞=1,  then
(4.3)∥Pφg∥e,𝒟2→ℬμ=lim sup|φ(z)|→1μ(z)|g(z)|(lne1-|φ(z)|2)1/2.Proof.
SincePφg:𝒟2→ℬμ is bounded, for the test function f(z)≡1,  we get g∈Hμ∞. If ∥φ∥∞<1, then from Lemma 2.3 it follows that Pφg:𝒟2→ℬμ is compact which is equivalent with ∥Pφg∥e,𝒟2→ℬμ=0. On the other hand, it is clear that in this case the condition |φ(z)|→1 is vacuous, so that (4.3) is vacuously satisfied.
Now assume that∥φ∥∞=1,  and that (φ(zk))k∈ℕ is a sequence in 𝔹 such that |φ(zk)|→1 as k→∞. For w∈𝔹 fixed, set
(4.4)fw(z)=ln(e/(1-〈z,w〉))(ln(e/(1-|w|2)))1/2,z∈𝔹.
By (1.9), we have that ∥fw∥𝒟2=1, for each w∈𝔹. Hence, the sequence (fφ(zk))k∈ℕ is such that ∥fφ(zk)∥𝒟2=1, for each k∈ℕ, and clearly it converges to zero uniformly on compacts of 𝔹. From this and by  [35, Theorems 6.19], it easily follows that fφ(zk)→0 weakly in 𝒟2, as k→∞. Hence, for every compact operator K:𝒟2→ℬμ, we have that
(4.5)limk→∞∥Kfφ(zk)∥ℬμ=0.
Thus, for every such sequence and for every compact operator K:𝒟2→ℬμ,  we have that
(4.6)∥Pφg+K∥𝒟2→ℬμ≥lim supk→∞∥Pφgfφ(zk)∥ℬμ-∥Kfφ(zk)∥ℬμ∥fφ(zk)∥𝒟2=lim supk→∞∥Pφgfφ(zk)∥ℬμ≥lim supk→∞μ(zk)|g(zk)fφ(zk)(φ(zk))|=lim supn→∞μ(zk)|g(zk)|(lne1-|φ(zk)|2)1/2.
Taking the infimum in (4.6) over the set of all compact operators K:𝒟2→ℬμ,  we obtain
(4.7)∥Pφg∥e,𝒟2→ℬμ≥lim supn→∞μ(zk)|g(zk)|(lne1-|φ(zk)|2)1/2,
from which an inequality in (4.3) follows.
In the sequel, we prove the reverse inequality. Assume that(rl)l∈ℕ is a sequence of positive numbers which increasingly converges to 1. Consider the operators defined by
(4.8)(Prlφgf)(z)=∫01f(rlφ(tz))g(tz)dtt,l∈ℕ.
Since ∥rlφ∥∞<1, by Lemma 2.3, we have that these operators are compact.
SincePφg:𝒟2→ℬμ is bounded, then g∈Hμ∞. Let ρ∈(0,1)  be fixed for a moment. By Lemma 2.1, we get
(4.9)∥Pφg-Prlφg∥𝒟2→ℬμ=sup∥f∥𝒟2≤1supz∈𝔹μ(z)|g(z)||f(φ(z))-f(rlφ(z))|≤sup∥f∥𝒟2≤1sup|φ(z)|≤ρμ(z)|g(z)||f(φ(z))-f(rlφ(z))|+sup∥f∥𝒟2≤1sup|φ(z)|>ρμ(z)|g(z)||f(φ(z))-f(rlφ(z))|≤∥g∥Hμ∞sup∥f∥𝒟2≤1sup|φ(z)|≤ρ|f(φ(z))-f(rlφ(z))|(4.10)+sup∥f∥𝒟2≤1sup|φ(z)|>ρμ(z)|g(z)||f(φ(z))-f(rlφ(z))|.
Further, we have(4.11)∥f-fr∥𝒟22=∑α|α|α!|α|!|aα|2(1-r|α|)2≤∑α|α|α!|α|!|aα|2≤∥f∥𝒟2.
From (1.10), (4.11),  and the fact f(z)-f(rz)∈𝒟2, we obtain
(4.12)|f(z)-f(rz)|≤∥f∥𝒟2(lne1-|z|2)1/2,
in particular
(4.13)|f(φ(z))-f(rlφ(z))|≤∥f∥𝒟2(lne1-|φ(z)|2)1/2.
Let(4.14)Il:=sup∥f∥𝒟2≤1sup|φ(z)|≤ρ|f(φ(z))-f(rlφ(z))|.
The mean value theorem along with the subharmonicity of the moduli of partial derivatives of f, well-known estimates among the partial derivatives of analytic functions, Theorem 6.2, and Proposition 6.2 in [35], yield
(4.15)Il≤sup∥f∥𝒟2≤1sup|φ(z)|≤ρ(1-rl)|φ(z)|sup|w|≤ρ|∇f(w)|≤Cρ(1-rl)sup∥f∥𝒟2≤1(∑j=1[n/p]|∇jf(0)|+sup|w|≤(1+ρ)/2|∇[n/p]+1f(w)|)≤Cρ(1-rl)sup∥f∥𝒟2≤1(∑j=1[n/p]|∇jf(0)|+(∫|w|≤(3+ρ)/4|∇[n/p]+1f(w)|2(1-|w|2)2([n/p]+1)dτ(w))1/2)≤Cρ(1-rl)sup∥f∥𝒟2≤1(∑j=1[n/p]|∇jf(0)|+∫𝔹|∇[n/p]+1f(w)|2(1-|w|2)2([n/p]+1)dτ(w))1/2≤Cρ(1-rl)→0,asl→∞,
where dτ(z)=dV(z)/(1-|z|2)n+1 and dV(z) is the Lebesgue volume measure on 𝔹.
Using (4.13) in (4.10), letting l→∞ in (4.9), using (4.15), and then letting ρ→1,  the reverse inequality follows, finishing the proof of the theorem.



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*Source: 134969-2010-11-04.xml*