Applied Mathematics and Computation 217 (2011) 6192–6197

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Essential norm of composition operators between weighted Hardy spaces

Stevo Stevic a,⇑, Ajay K. Sharma b

a Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbiab School of Mathematics, Shri Mata Vaishno Devi University, Kakryal, Katra 182320, J&K, India

a r t i c l e i n f o

Keywords:Composition operatorWeighted Hardy spaceBoundednessEssential normUnit disk

0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.12.103

⇑ Corresponding author.E-mail addresses: sstevic@ptt.rs (S. Stevic), aksju

a b s t r a c t

An asymptotic formula for the essential norm of composition operators acting betweentwo weighted Hardy spaces Hw1 and Hw2 , where w1 and w2 are two admissible weight func-tions, is given. The boundedness of the operators is also characterized.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

Let D be the open unit disk in the complex plane, D(z,r) the open unit disk centered at z with radius r, and HðDÞ the spaceof holomorphic functions on D. Throughout this paper constants are denoted by C, they are positive and not necessarily thesame at each occurrence. The notation A � B means that there is a positive constant C not depending on variables in A and Bsuch that B/C 6 A 6 CB.

For a 2 D, let ga be the involutive Möbius transformation of the unit disk, interchanging points a and 0, that is

gaðzÞ ¼a� z

1� �az:

Let w 2 C2[0,1) be a positive integrable function. If we extend it on D by wðzÞ ¼ wðjzjÞ; z 2 D , we call it a weight function. ByHw we denote the weighted Hardy space consisting of all f 2 HðDÞ such that

kfk2Hw¼ jf ð0Þj2 þ

ZD

jf 0ðzÞj2wðzÞdAðzÞ <1; ð1Þ

where dAðzÞ ¼ 1p dxdy ¼ 1

p rdrdh stands for the normalized area measure in D. A simple computation shows that a functionf ðzÞ ¼

P1n¼0anzn belongs to Hw if and only if

X1n¼0

wnjanj2 <1; ð2Þ

where w0 = 1 and

wn ¼ 2n2Z 1

0r2n�1wðrÞdr; n 2 N:

The sequence ðwnÞn2N0is called the weight sequence of the weighted Hardy space Hw. The properties of the weighted Hardy

spaces with the weight sequence ðwnÞn2N0, clearly depends upon wn. If w1n and w2n are correspondingly weight sequences of

. All rights reserved.

_76@yahoo.com (A.K. Sharma).

S. Stevic, A.K. Sharma / Applied Mathematics and Computation 217 (2011) 6192–6197 6193

two weighted Hardy spaces Hw1 and Hw2 , respectively and w1n � w2n, then from (2) we have that f 2 Hw1 if and only iff 2 Hw2 .

For some other weights, weighted-type spaces and operators on them see, e.g. [1,2,4,5,14,25,30] and the referencestherein.

Let ka(r) = (1 � r2)a, a > �1. Then the Hardy space H2 can be identified with Hk1, with the weight sequence wn ¼ 1; n 2 N0.

The Dirichlet space Da; a 2 ½0;1Þ, is exactly Hka . For the case a = 0 the weight sequence is w0 ¼ 1; wn ¼ n; n 2 N. Theweighted Bergman space

A2a ¼ f 2 HðDÞ

ZD

jf ðzÞj2ð1� jzj2ÞadAðzÞ <1����

� �;

where a > �1, can be identified with Hkaþ2 .Throughout the paper, a weight w will satisfy some of the following properties:

ðW1Þ w is non-increasing;ðW2Þ wðrÞ

ð1�rÞ1þd is nondecreasing for some d > 0;

ðW3Þ limr!1�wðrÞ ¼ 0.

One of the next two properties of convexity is fulfilled:

ðW4Þ w is convex and limr!1�w0ðrÞ ¼ 0;

ðW5Þ w is concave.

Such a weight function is called admissible ([15]). If w satisfies conditions: ðW1Þ; ðW2Þ; ðW3Þ and ðW4Þ, then it is said thatw is (I)-admissible. If w satisfies ðW1Þ; ðW2Þ; ðW3Þ and ðW5Þ, then it is said that w is (II)-admissible. (I)-admissibility corre-sponds to the case H2

(Hw � A2a for some a > �1, whereas (II)-admissibility corresponds to the case D(Hw � H2. If we say that

a weight is admissible it means that it is (I)-admissible or (II)-admissible.Let u be a holomorphic self-map on D. The composition operator Cu induced by u is defined by Cuf = f�u for f 2 HðDÞ.

These operators have gained increasing attention during the last three decades, mainly due to the fact that they provide alink between classical function theory and, functional analysis and operator theory (see, e.g. [2,6–18,20–22,25,28–35]).

Recall that the essential norm kTke of a bounded linear operator T between Banach spaces X and Y is given by

kTke ¼ inffkT þ KkX!Y : K : X ! Y is compactg:

It provides a measure of non-compactness of T. Clearly, T is compact if and only if kTke = 0.Let w1 and w2 be two admissible weight functions. Motivated by [15], in this paper we study the composition operator

Cu : Hw1 ! Hw2 . The boundedness of the operator is characterized first, which slightly extends Theorem 1.3 in [15]. In themain result of this paper an asymptotic formula for the essential norm of the operator is given which considerably extendsthe main result in [15].

For some recent results regarding calculation of norms and estimating essential norms of concrete linear operators be-tween spaces of holomorphic functions on various domains see, e.g. [2–4,6,7,9,10,16–23,25–31] and the related referencestherein.

2. Auxiliary results

Let u be a holomorphic self-map of D. The generalized Nevanlinna counting function associated to a weight function w isdefined for every z 2 D n fuð0Þg by

Nu;wðzÞ ¼X

uðkÞ¼z

wðkÞ;

where, by convention, Nu;wðzÞ ¼ 0 when z = u(0). Note that Nu;wðzÞ ¼ 0 when z R uðDÞ.Counting functions play an important role in characterizing bounded and compact composition operators on different

spaces of analytic functions, see, e.g. [2,10–13,15,24] and the related references therein.The following non-univalent change of variable formula is a well-known result which essentially stems from measure

theory (see, e.g. Theorem 2.32 in [2]).

Lemma 1. If g and w are positive measurable functions on D and u is a holomorphic self-map of D, then

ZD

ðg �uÞðzÞju0ðzÞj2wðzÞdAðzÞ ¼Z

D

gðzÞNu;wðzÞdAðzÞ: ð3Þ

The next two lemmas are from [15] and they will be important in the proofs of the main results of the paper.

6194 S. Stevic, A.K. Sharma / Applied Mathematics and Computation 217 (2011) 6192–6197

Lemma 2 [15, Lemmas 2.2 and 2.3]. Let w be an admissible weight and u be a holomorphic self-map of D. Then the generalizedNevanlinna counting function Nu;w satisfies the next sub-mean value property: for every r > 0 and every z 2 D such thatDðz; rÞ � D n Dð0;1=2Þ

Nu;wðzÞ 62r2

ZDðz;rÞ

Nu;wðfÞdAðfÞ: ð4Þ

Lemma 3 [15, Lemma 2.4]. Let d > 0 and w be a weight satisfying ðW1Þ and ðW2Þ , then

ZD

wðzÞj1� �kzj4þ2d

dAðzÞ � wðkÞð1� jkj2Þ2þ2d

:

Also, if

fkðzÞ ¼1ffiffiffiffiffiffiffiffiffiffiwðkÞ

p ð1� jkj2Þ1þd

ð1� �kzÞ1þd; ð5Þ

then

kfkkHw� 1:

Let Rn be the orthogonal projection of Hw onto znHw and Qn = I � Rn, that is, for f ¼P1

k¼0akzk 2 Hw , let

ðRnf ÞðzÞ ¼X1k¼n

akzk

and

ðQ nf ÞðzÞ ¼Xn�1

k¼0

akzk:

The following estimates will also play an important role in this paper.

Lemma 4 [2, Proposition 3.15]. Let Hw be a weighted Hardy space with the weight sequence ðwnÞn2N0. Then for each r 2 (0,1)

and f 2 Hw

(a) jðRnf ÞðzÞj 6 kfkHw

P1k¼n

r2k

w2k

� �1=2

for jzj 6 r;

(b) jðRnf Þ0ðzÞj 6 kfkHw

P1k¼nk2 r2ðk�1Þ

w2k

� �1=2

for jzj 6 r ,

where wk ¼ kzkkHw; k 2 N0.

Lemma 5. Let u be a holomorphic selfmap of D and the operator Cu : Hw1 ! Hw2 is bounded. Then

kCuke 6 lim infn!1

kCuRnkHw1!Hw2: ð6Þ

Proof. Since Rn + Qn = I and Qn is a compact operator on Hw1 , we have that for each n 2 N

kCuke ¼ kCuRn þ CuQnke 6 kCuRnke 6 kCuRnkHw1!Hw2;

from which inequality (6) follows. h

3. Main results

In this section we formulate and prove the main results of this paper. The first theorem is an extension of Theorem 1.3 in[15].

Theorem 1. Let w1 and w2 be admissible weights, u be a holomorphic self-map of D. Then Cu : Hw1 ! Hw2 is bounded if and onlyif

M :¼ supz2D

Nu;w2 ðzÞw1ðzÞ

<1: ð7Þ

S. Stevic, A.K. Sharma / Applied Mathematics and Computation 217 (2011) 6192–6197 6195

Moreover, if Cu : Hw1 ! Hw2 is bounded, then

kCuk2Hw1!Hw2

� M: ð8Þ

Proof. Suppose that (7) holds. Then using Lemma 1, the fact that the point evaluation operator is continuous on weightedHardy spaces and condition (7) we obtain

kCufk2Hw2¼ jf ðuð0ÞÞj2 þ

ZD

jf 0ðuðzÞÞj2ju0ðzÞj2w2ðzÞdAðzÞ ¼ jf ðuð0ÞÞj2 þZ

uðDÞjf 0ðzÞj2Nu;w2 ðzÞdAðzÞ

6 Ckfk2Hw1þM

ZD

jf 0ðzÞj2w1ðzÞdAðzÞ � CMkfk2Hw1

; ð9Þ

from which the boundedness of Cu : Hw1 ! Hw2 follows.Next assume that Cu : Hw1 ! Hw2 is bounded. Let fk be the test function in (5) with w = w1. Then by the asymptotic

relation

j1� �kzj � 1� jkj2; z 2 Dðk; ð1� jkjÞ=2Þ;

and Lemma 2 we have that

kCufkk2Hw2¼ jfkðuð0ÞÞj2 þ

ZD

jf 0kðuðzÞÞj2ju0ðzÞj2w2ðzÞdAðzÞP jkj2ð1þ dÞ2 ð1� jkj

2Þ2þ2d

w1ðkÞ

ZD

ju0ðzÞj2w2ðzÞj1� �kuðzÞj4þ2d

dAðzÞ

¼ jkj2ð1þ dÞ2 ð1� jkj2Þ2þ2d

w1ðkÞ

ZD

Nu;w2 ðzÞj1� �kzj4þ2d

dAðzÞP jkj2ð1þ dÞ2 ð1� jkj2Þ2þ2d

w1ðkÞ

ZD k;1�jkj2ð Þ

Nu;w2 ðzÞj1� �kzj4þ2d

dAðzÞ

P C1

w1ðkÞ1

ð1� jkj2Þ2Z

D k;1�jkj2ð ÞNu;w2 ðzÞdAðzÞP C

Nu;w2 ðkÞw1ðkÞ

;

for jkj close to 1, say jkj > 1 � d1.On the other hand, it is easy to see that there is a C > 0 such that

supjkj61�d1

Nu;w2 ðkÞw1ðkÞ

6 CkCuk2Hw1!Hw2

:

Thus we have

supz2D

Nu;w2 ðzÞw1ðzÞ

6 C supk2DkCufkk2

Hw26 CkCuk2

Hw1!Hw2; ð10Þ

which is bounded. From (9) and (10) we get (8), finishing the proof of the theorem. h

The next theorem estimates the essential norm of the operator Cu : Hw1 ! Hw2 . We are partially motivated by Chapter 3.2in [2].

Theorem 2. Let w1 and w2 be admissible weights and u be a holomorphic self-map of D. Suppose that Cu : Hw1 ! Hw2 is bounded.Then

kCuk2e � lim sup

jzj!1

Nu;w2 ðzÞw1ðzÞ

:

Proof. Upper bound. By Lemma 5, we have

kCuke 6 lim infn!1

supkfkHw1 61

kCuRnfkHw2:

Let kfkHw16 1. We have

kCuRnfk2Hw2¼ jðRnf �uÞð0Þj2 þ

ZD

jðRnf �uÞ0ðzÞj2w2ðzÞdAðzÞ: ð11Þ

By Lemma 4 (a) and since kfkHw16 1, it follows that

jðRnf �uÞð0Þj2 6X1k¼n

juð0Þj2k

w21k

! 0; as n!1:

6196 S. Stevic, A.K. Sharma / Applied Mathematics and Computation 217 (2011) 6192–6197

By Lemma 1, the second term in (11) is

I :¼Z

uðDÞjðRnf Þ0ðzÞj2Nu;w2 ðzÞdAðzÞ: ð12Þ

Fix an r 2 (0,1). We have

I 6 supz2rDjðRnf Þ0ðzÞj2

ZuðDÞ

Nu;w2 ðzÞdAðzÞ þZ

uðDÞnrDjðRnf Þ0ðzÞj2Nu;w2 ðzÞdAðzÞ; ð13Þ

where rD ¼ fz 2 D : jzj 6 rg.Applying Lemmas 1 and 4 (b) we obtain

Jn :¼ supz2rDjðRnf Þ0ðzÞj2

ZuðDÞ

Nu;w2 ðzÞdAðzÞ 6 kfk2Hw1

X1k¼n

k2

w21k

r2k�2

!kuk2

Hw2: ð14Þ

Since Cu : Hw1 ! Hw2 is bounded, then for f ðzÞ � z 2 Hw1 we obtain u 2 Hw2 . Using this fact and letting n ?1 in (14) we getlimn?1Jn = 0.

Further, we have

ZuðDÞnrD

jðRnf Þ0ðzÞj2Nu;w2 ðzÞdAðzÞ 6Z

uðDÞnrDjf 0ðzÞj2Nu;w2 ðzÞdAðzÞ ¼: L: ð15Þ

Set

Mr ¼ supr<jzj<1

Nu;w2 ðzÞw1ðzÞ

:

Then

L 6 Mr

ZuðDÞnrD

jf 0ðzÞj2w1ðzÞdAðzÞ 6 Mrkfk2Hw16 Mr :

Thus, we have

kCuk2e 6 Mr :

Letting in the last inequality r ? 1, we get the desired upper bound.Lower bound. Let ðanÞn2N be a sequence in D such that janj? 1 as n ?1 and fan be as in (5) with k = an. We know

kfankHw1� 1. Moreover, by ðW2Þ we have

ð1� janj2Þ1þdffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw1ðanÞ

p 6ð1� janj2Þð1þdÞ=2ffiffiffiffiffiffiffiffiffiffiffiffiffi

w1ð0Þp

which implies that fan converges to 0 uniformly on compacts of D as n ?1. Therefore fan tends weakly to zero as n ?1.Hence for any compact operator K : Hw1 ! Hw2 we have kKfankHw2

! 0 as n ?1. Therefore,

kCu � KkHw1!Hw2P lim sup

n!1kðCu � KÞfankHw2

P lim supn!1

kCufankHw2� kKfankHw2

� ¼ lim sup

n!1kCufankHw2

: ð16Þ

Taking the infimum in (16) over all compact operators K : Hw1 ! Hw2 we get

kCuk2e P lim sup

n!1kCufank

2Hw2¼ lim sup

n!1jðfan �uÞð0Þj

2 þZ

D

jðfan �uÞ0ðzÞj2w2ðzÞdAðzÞ

� �

¼ lim supn!1

ZD

jf 0anðzÞj2Nu;w2 ðzÞdAðzÞ; ð17Þ

where we have used the fact that jðfan �uÞð0Þj ! 0 as n ?1. We have

ZD

jf 0anðzÞj2Nu;w2 ðzÞdAðzÞ ¼ ð1þ dÞ2janj2

w1ðanÞ

ZD

ð1� janj2Þ2dþ2

j1� �anzj4þ2dNu;w2 ðzÞdAðzÞ

Pð1þ dÞ2janj2

w1ðanÞ

ZD an ;

1�jan j2ð Þð1� janj2Þ2dþ2

j1� �anzj4þ2dNu;w2 ðzÞdAðzÞ

P Cð1þ dÞ2janj2

w1ðanÞð1� janj2Þ2Z

D an ;1�jan j

2ð ÞNu;w2 ðzÞdAðzÞP Cð1þ dÞ2janj2

Nu;w2 ðanÞw1ðanÞ

;

where in the last inequality we have used Lemma 2.

S. Stevic, A.K. Sharma / Applied Mathematics and Computation 217 (2011) 6192–6197 6197

Therefore

lim supn!1

ZD

jf 0anðzÞj2Nu;w2 ðzÞdAðzÞP Cð1þ dÞ2 lim sup

n!1

Nu;w2 ðanÞw1ðanÞ

: ð18Þ

Combining (17) and (18), and since ðanÞn2N is an arbitrary sequence whose modulus tend to one, we obtain

kCuk2e P C lim sup

jzj!1

Nu;w2 ðzÞw1ðzÞ

;

finishing the proof of the theorem. h

Corollary 1. Let w1 and w2 be admissible weights ((I)-admissible or (II)-admissible), and u be a holomorphic self-map of D. Sup-pose that Cu : Hw1 ! Hw2 is bounded. Then Cu : Hw1 ! Hw2 is compact if and only if

lim supjzj!1

Nu;w2 ðzÞw1ðzÞ

¼ 0:

Acknowledgments

The work of the second author is partially supported by National Board of Higher Mathematics (NBHM)/DAE, India (GrantNo. 48/4/ 2009/R&D-II/426).

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