On the Norm of Certain Weighted Composition Operators on the Hardy Space

Abstract

We obtain a representation for the norm of certain compact weighted composition operator _{Cψ,[straight phi]} on the Hardy space ^H2 , whenever [straight phi](z)=az+b and ψ(z)=az-b . We also estimate the norm and essential norm of a class of noncompact weighted composition operators under certain conditions on [straight phi] and ψ . Moreover, we characterize the norm and essential norm of such operators in a special case.

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Recommended by Stevo Stevic

Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, Iran

Received 22 January 2009; Revised 9 March 2009; Accepted 8 May 2009

1. Introduction

Let D denote the open unit disk in the complex plane. The Hardy space ^H2 is the space of analytic functions on D whose Taylor coefficients, in the expansion about the origin, are square summable. Also we recall that ^H∞ is the space of all bounded analytic function defined on D . For α∈D , the reproducing kernel at α for ^H2 is defined by _Kα (z)=1/(1-α...z). An easy computation shows that ...f,_Kα ...=f(α) whenever f∈^H2 . For any analytic self-map [straight phi] of D , the composition operator _{C[straight phi]} on ^H2 is defined by the rule _{C[straight phi]} (f)=f[composite function][straight phi] . Every composition operator is bounded, with

[figure omitted; refer to PDF] (see [1]). We see from expression (1.1) that ||_{C[straight phi]} ||=1 whenever [straight phi](0)=0 . There are few other cases for which the exact value of the norm has been known for many years. For example, the norm of _{C[straight phi]} was obtained by Nordgren in [2], whenever [straight phi] is an inner function. In [3] this norm was determined, when [straight phi](z)=az+b , with |a|+|b|≤1, and if 0<s<1 and 0≤r≤1 the norm was found in [4] for [straight phi](z)=((r+s)z+(1-s))/(r(1-s)z+(1+rs)) .

In 2003, Hammond [5] obtained exact values for the norms of composition operators _{C[straight phi]} for certain linear fractional maps [straight phi] . In [6], Bourdon et al. determined the norm of _{C[straight phi]} for a large class of linear-fractional maps, including those of the form [straight phi](z)=b/(d-z) , where 0<b<d-1. The connection between the norm of certain composition operators _{C[straight phi]} with linear-fractional symbol acting on the Hardy space and the roots of associated hypergeometric functions was first made by Basor and Retsek [7]. It was later refined by Hammond [8]. In [9] Effinger-Dean et al. computed the norms of composition operators with rational symbols that satisfy certain properties. Their work is based on the initial work of Hammond [5]. Some other recent results regarding the calculation of the operator norm of some composition operators on the other spaces can be found in [10-14].

If ψ is a bounded analytic function on D and [straight phi] is an analytic map from D into itself, the weighted composition operator _{Cψ,[straight phi]} is defined by _{Cψ,[straight phi]} (f)(z)=ψ(z)f([straight phi](z)) . The map [straight phi] is called the composition map and ψ is called the weight. If ψ is a bounded analytic function on D , then the operator can be rewritten as _{Cψ,[straight phi]} =_{MψC[straight phi]} , where _Mψ is a multiplication operator and _{C[straight phi]} is a composition operator. Recall that if [straight phi] is an analytic self-map of D , then the composition operator _{C[straight phi]} on ^H2 is bounded, hence in this case _{Cψ,[straight phi]} is bounded, but in general every weighted composition operator _{Cψ,[straight phi]} on ^H2 is not bounded. If _{Cψ,[straight phi]} is bounded, then _{Cψ,[straight phi]} (1)=ψ belongs to ^H2 . These operators come up naturally. In 1964, Forelli [15] showed that every isometry on ^Hp for 1<p<∞ and p≠2 is a weighted composition operator. Recently there has been a great interest in studying weighted composition operators in the unit disk, polydisk, or the unit ball; see [12, 16-27], and the references therein. In this paper we investigate the norm of certain bounded weighted composition operators _{Cψ,[straight phi]} on ^H2 .

2. Norm Calculation

In this section we obtain a representation for the norm of a class of compact weighted composition operators _{Cψ,[straight phi]} on the Hardy space ^H2 , whenever [straight phi](z)=az+b , ψ(z)=az-b , ^|b|2 ≥1/2, and 2^|a|2 +^|b|2 ≤2/3 . Also we give the norm and essential norm inequality for a class of noncompact weighted composition operators _{Cψ,[straight phi]} on ^H2 when [straight phi](z)=a^zn +b , for some n∈... , |a|+|b|=1, and ψ is a bounded analytic map on D such that the radial limit of |ψ| at one of the n th roots of b|a|/a|b| is the supremum of |ψ| on D . Also, when n=1 we obtain the norm and essential norm of such operators.

The following lemma was inspired by a similar result for unweighted composition operators [28, Theorem 1.4]. See [29] for a similar proof.

Lemma 2.1.

Let _Kw be the reproducing kernel at w . Then [figure omitted; refer to PDF] In the next proposition we generalize the lower bound in (1.1).

Proposition 2.2.

Let [straight phi] be a nonconstant analytic self-map of D , and let ψ be a nonzero analytic map on D . If n is the smallest nonnegative integer such that ^ψ(n) (0)≠0 , then [figure omitted; refer to PDF]

Proof.

We note that if f is in ^H2 , then for every n∈...∪{0} we have |^f(n) (0)/n!|≤_||f||2 . Hence we have [figure omitted; refer to PDF]

Let T be a bounded operator on a Hilbert space H . We recall that _||T||e , the essential norm of T , is the norm of its equivalence class in the Calkin algebra. Since the spectral radius of the operator ^T* T equals ||^T* T||=^||T||2 , we study the spectrum of ^T* T when trying to determine ||T|| . We say that the operator T is norm-attaining if there is a nonzero h∈H such that ||T(h)||=||T||||h||. We know that ||T(h)||=||T||||h|| if and only if ^T* T(h)=^||T||2 h . Moreover, if _||T||e <||T|| , then the operator T is norm-attaining and so the quantity ^||T||2 equals the largest eigenvalue of ^T* T ; see [5] for more details. If [straight phi](z)=az+b , ψ(z)=az-b , ^|b|2 ≥1/2, and 2^|a|2 +^|b|2 ≤2/3 , then the operator _{Cψ,[straight phi]} is compact (see the proof of Proposition 2.5). Hence 0=_{||Cψ,[straight phi] ||e} <||_{Cψ,[straight phi]} || and so _{Cψ,[straight phi]} is norm-attaining.

Now our goal is to find a functional equation that relates an eigenvalue of ^{Cψ,[straight phi]*}_{Cψ,[straight phi]} to the values of its eigenfunctions at particular points in the disk. In what follows we use the techniques used in [5, 6, 30] and present some results that help us to obtain the norm of _{Cψ,[straight phi]} .

Let [straight phi] be an analytic self-map of D and let ψ be a bounded analytic map on D . Then [figure omitted; refer to PDF]

But if [straight phi](z)=az+b such that |a|+|b|≤1 , then by [3] or [28] [figure omitted; refer to PDF]

where h(z)=1 , g(z)=1/-b¯z+1, and σ(z)=a¯z/-b¯z+1.

From now on, unless otherwise stated, we assume that ψ(z)=cz+d, [straight phi](z)=az+b, and |a|+|b|≤1 . Since ^Tz* is the backward shift on ^H2 , we see that [figure omitted; refer to PDF] for all z in D not equal to 0, where [figure omitted; refer to PDF]

In particular, if g is an eigenfunction for ^{Cψ,[straight phi]*}_{Cψ,[straight phi]} corresponding to an eigenvalue λ , then [figure omitted; refer to PDF] Formula (2.8) is essentially identical to [5, Formula (3.3)]. Using (2.8) we can find a set of conditions under which we determine ||^{Cψ,[straight phi]*}_{Cψ,[straight phi]} || . In the trivial case a=0 we have ||_{Cψ,[straight phi]} ||=_||ψ||2 (1/1-^|b|2 ). Also if d=0 , then ||_{Cψ,[straight phi]} ||=|c|||_{C[straight phi]} || and if c=0 , then ||_{Cψ,[straight phi]} ||=|d|||_{C[straight phi]} || . Therefore we assume that a,b,c,d are nonzero.

Throughout this paper, we write ^τ[j] to denote the j th iterate of τ, that is, ^τ[0] is the identity map on D and ^τ[j+1] =τ[composite function]^τ[j] .

By a similar argument as in the proof of [5, Proposition 5.1], we have the following lemma.

Lemma 2.3.

Let g be an eigenfunction for ^{Cψ,[straight phi]*}_{Cψ,[straight phi]} corresponding to an eigenvalue λ , z∈D and for each nonnegative integer j , ^τ[j] (z)≠0 . Then one has [figure omitted; refer to PDF] where one takes ^∏m=0-1 (·)=1.

Lemma 2.4.

For each n∈... , ^τ[n] (0)=_αn b , where {_αn } is strictly increasing sequence such that _αn ≥1 for each n∈... . Also _αn+1 =1+_αn^|a|2 /(1-_αn^|b|2 ).

Proof.

(By induction) Since τ(0)=b and ^τ[2] (0)=(1+^|a|2 /(1-^|b|2 ))b , the claim holds for n=1 . Assume the claim holds for n-1 . We will prove it for n . We have [figure omitted; refer to PDF] Now if we set _αn =1+(_αn-1^|a|2 )/(1-_αn-1^|b|2 ) , then ^τ[n] (0)=_αn b . But by hypothesis _αn-1 <_αn , so [figure omitted; refer to PDF] which implies that _αn <_αn+1 also ^τ[n+1] (0)=τ(_αn b)=(1+_αn^|a|2 /(1-_αn^|b|2 ))b. Hence the proof is complete.

Proposition 2.5.

Let a=c , b=-d and let λ=^{||_{Cψ,[straight phi]} ||2} . If ^|b|2 ≥1/2, and 2^|a|2 +^|b|2 ≤2/3 , then for each z∈D with the property that ^τ[j] (z)≠0 for every nonnegative integer j , one has [figure omitted; refer to PDF]

Proof.

Since 2^|a|2 +^|b|2 ≤2/3 , it is easy to see that |a|+|b|=1 if and only if |a|=1/3 and |b|=2/3 . By assumption ^|b|2 ≥1/2 , so |a|+|b|<1 . Therefore _{C[straight phi]} is compact and, since _{Cψ,[straight phi]} =_{MψC[straight phi]} , the operator _{Cψ,[straight phi]} is compact. Now according to the paragraph after Proposition 2.2, there is function g in ^H2 such that ^{Cψ,[straight phi]*}_{Cψ,[straight phi]} g=λg . Let z∈D and for each integer j≥0 , ^τ[j] (z)≠0 . By Lemma 2.3, we have [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF] Now if _w0 is the Denjoy-Wolff point of τ, it suffices to show that [figure omitted; refer to PDF] Suppose the above inequality holds. Then we conclude that there is 0<β<1 and N∈... such that for k>N we have |γ(^τ[k] (z))/λ|<β<1 . Now we break the proof into two parts.

(1) The Denjoy-Wolff point _w0 of τ lies inside D , then g(^τ[j] (z)) converges to g(_w0 ) . Hence

[figure omitted; refer to PDF]

(2) The Denjoy-Wolff point _w0 of τ lies on ∂D , then by [31, Lemma 5.1] τ must be parabolic and by [6, Lemma 3.3] there is a constant C such that

[figure omitted; refer to PDF] Thus it follows that [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF] Now we show that |γ(_w0 )/λ|<1 . Since a=c and b=-d , we see that [figure omitted; refer to PDF] By [30], we have [figure omitted; refer to PDF] Applying the assumptions ^|b|2 ≥1/2 and 2^|a|2 +^|b|2 ≤2/3 , an easy computation shows that [figure omitted; refer to PDF] Also by using Proposition 2.2, 1/λ<(1-^|b|2 )/^|b|2 , and by Lemma 2.4, there is _αn ≥1 such that ^τ[n] (0)=_αn b . Therefore [figure omitted; refer to PDF]

Proposition 2.6.

Let a=c , b=-d , ^|b|2 ≥1/2, and 2^|a|2 +^|b|2 ≤2/3 . Then λ=^{||_{Cψ,[straight phi]} ||2} satisfies the equation [figure omitted; refer to PDF]

Proof.

Since for every integer j≥0 , ^τ[k] ([straight phi](0))≠0 , in Proposition 2.5 we set z=[straight phi](0) , then we have [figure omitted; refer to PDF] Since [straight phi](0)=τ(0) , we see that [figure omitted; refer to PDF] But g([straight phi](0))≠0 , because otherwise Proposition 2.5 would dictate that the function g(z) is identically 0. Thus eigenfunction g must have the property that g([straight phi](0))≠0 . Hence we have [figure omitted; refer to PDF]

We define [figure omitted; refer to PDF] Now we characterize the properties of F and by using these properties we obtain a formula for the norm of _{Cψ,[straight phi]} . The idea behind Proposition 2.7 is similar to the one found in [30].

Proposition 2.7.

Let a=c , b=-d , ^|b|2 ≥1/2, and 2^|a|2 +^|b|2 ≤2/3 . Then F(z) has the following properties.

(a) The power series that defines F(z) has radius of convergence _r0 larger than 1/λ .

(b) F(x) is non-negative real number for all x in the interval [0,_r0 ) .

(c) ^{F[variant prime]} (x)>0 for all x in the interval (0,_r0 ) .

Proof.

(a) By Lemma 2.4, for each positive integer n there is _αn ≥1 such that ^τ[n] (0)=_αn b , then χ(^τ[m+1] (0))=1/_αm+1 ≤1 . Also in the proof of Proposition 2.5 we have |γ(_w0 )/λ|<1 , hence there is 0<β<1 and N∈... such that if n>N , then [figure omitted; refer to PDF] Now let β<_β1 <1 and 0<...<λ(_β1 -β)/_β1 . Then if n>N we have [figure omitted; refer to PDF] Therefore there is a constant C such that [figure omitted; refer to PDF] By Lemma 2.4, there is strictly increasing sequence _αn ≥1 such that ^τ[n] (0)=_αn b , and by hypothesis |b|>2/2 , hence 1-2_αn^|b|2 <1-2^|b|2 <0 . Also we have ^|a|2 +^|b|2 ≤|b|≤|b/_w0 |<1/_αn , so we conclude that -(1-_αn^|b|2 )+^|a|2_αn <0 . Therefore

[figure omitted; refer to PDF] Also it is obvious that [figure omitted; refer to PDF] Hence the proof of part (b) is complete.

(c) Every coefficient of F is positive and so ^{F[variant prime]} (x)>0 for all x in the interval (0,_r0 ) .

Now we find an equation that involves the norm of _{Cψ,[straight phi]} .

Theorem 2.8.

Let a=c , b=-d , ^|b|2 ≥1/2 and 2^|a|2 +^|b|2 ≤2/3 . Then λ=^{||_{Cψ,[straight phi]} ||2} is the unique positive real solution of the equation [figure omitted; refer to PDF]

Proof.

By Propositions 2.6 and 2.7, there is exactly one positive real number λ which satisfies equation (2.34), and this number must be equal to ^{||_{Cψ,[straight phi]} ||2} .

Corollary 2.9.

In Theorem 2.8 if one replaces _a0 with a and _b0 with b such that |a|=|_a0 | , and |b|=|_b0 | , then norm of _{Cψ,[straight phi]} does not change.

Proof.

We have ^τ[n] (0)=_αn b . But by Lemma 2.4, _αn =1+_αn-1^|a|2 /(1-_αn-1^|b|2 ) . Hence if one replaces _a0 with a and _b0 with b such that |a|=|_a0 | and |b|=|_b0 | , then _αn , γ(^τ[m+1] (0)) and χ(^τ[m+1] (0))=1/_αm+1 do not change. Hence by (2.34), the norm of _{Cψ,[straight phi]} does not change.

Example 2.10.

Let [straight phi](z)=az+b and ψ(z)=az-b , where |a|=1/10 and |b|=8/10 . Then we have [figure omitted; refer to PDF] For positive integer _k0 , let _λk0 denote the positive solution of [figure omitted; refer to PDF] Now by using numerical methods, we have [figure omitted; refer to PDF] Hence we see that ^{||_{Cψ,[straight phi]} ||2} [approximate]1.797084948.

The hypotheses of Theorem 2.8 restrict us to considering the norms of compact operators. In the remainder of this section we investigate the norm and essential norm of a class of noncompact weighted composition operators.

Theorem 2.11.

Let [straight phi](z)=a^zn +b , for some n∈... , where |a|+|b|=1 , ψ∈^H∞ ,let α be one of the n th roots of b|a|/a|b| such that ψ has radial limit at α, and let |ψ| attains its supremum on D∪{α} at α . Then [figure omitted; refer to PDF]

Proof.

Let 0<r<1 . Taking β=rα , by a similar proof for unweighted composition operators [28, Proposition 3.13], we have [figure omitted; refer to PDF] Therefore [figure omitted; refer to PDF] On the other hand, by [3], we have [figure omitted; refer to PDF] Therefore [figure omitted; refer to PDF]

Corollary 2.12.

In Theorem 2.11 if n=1 , then [figure omitted; refer to PDF]

Example 2.13.

(1) If [straight phi](z)=(1/2)z+1/2 and ψ(z)=(z+1)/2 , then ||_{Cψ,[straight phi]} ||=2.

(2) If [straight phi](z)=(1/3)z+(2/3)i and ψ(z)=^z5 -2^z3 +i , then ||_{Cψ,[straight phi]} ||=43.

(3) If [straight phi](z)=-(1/4)iz+3/4 and ψ(z)=(7^z5 -5^z3 +2i)/^(z2 +2) , then ||_{Cψ,[straight phi]} ||=28.

Acknowledgment

The authors would like to thank the referee for his valuable comments and suggestions.

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