- Email: [email protected]
Norm of weighted composition operators from α -Bloch spaces to weighted-type spaces
Applied Mathematics and Computation 215 (2009) 818–820
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Norm of weighted composition operators from a-Bloch spaces to weighted-type spaces Stevo Stevic´ Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia
a r t i c l e
i n f o
Keywords: Weighted composition operator a-Bloch space Weighted-type space Operator norm
a b s t r a c t We calculate in an elegant way operator norm of the weighted composition operator from the a-Bloch space, with a 2 ð0; 1Þ n f1g, to a weighted-type space on the unit ball. This result can be regarded as a complement to our recent result regarding the same problem for the case a ¼ 1: Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction and preliminaries Let Bn ¼ B be the open unit ball in the complex vector space Cn , B1 ¼ D the open unit disk in C, HðBÞ the class of all holomorphic functions on the unit ball and H1 ðXÞ the space of all bounded holomorphic functions on a set X. For an f 2 HðBÞ we denote by rf the gradient of the function f, that is
rf ¼
@f @f ;...; : @z1 @zn
The a-Bloch space Ba ¼ Ba ðBÞ, consists of all f 2 HðBÞ such that
ba ðf Þ ¼ supð1 jzjÞa jrf ðzÞj < 1: z2B
It is easy to see that the a-Bloch space with the norm
kf kBa ¼ jf ð0Þj þ ba ðf Þ becomes a Banach space. The little a-Bloch space Ba0 is a subspace of Ba consisting of all f such that
lim ð1 jzjÞa jrf ðzÞj ¼ 0:
jzj!1
1 The weighted-type space H1 l ¼ Hl ðBÞ consists of all f 2 HðBÞ such that
kf kH1l :¼ sup lðzÞjf ðzÞj < 1; z2B
where l is a weight, i.e., a positive continuous function on B. Assume u 2 HðBÞ and u is a holomorphic self-map of B: The weighted composition operator induced by u and u is defined on HðBÞ by
ðuC u f ÞðzÞ ¼ uðzÞf ðuðzÞÞ: E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.06.005
S. Stevic´ / Applied Mathematics and Computation 215 (2009) 818–820
819
A typical problem is to provide function theoretic characterizations when u and u induce bounded or compact weighted composition operators between two given spaces of holomorphic functions (see, for example, [2]). The boundedness and compactness of weighted composition operators between the Bloch space and H1 ðDÞ was studied in [10]. The case of the unit polydisk was studied in [6,7] (the case of composition operators, i.e., when uðzÞ 1, was treated in [12]). The same problems in the setting of the unit ball were investigated in [8]. For related results see also [1,3– 5,9,13,20,21,23,24] and the references therein. It is of some interest to calculate operator norm of weighted composition operators. For some recent related results, see, e.g., [2,15–19,22]. In [16], we calculated operator norm of the operator uC u : BðBÞðor B0 ðBÞÞ ! H1 l ðBÞ. We proved the following formula
1 1 þ juðzÞj kuC u kBðBÞðor B0 ðBÞÞ!H1l ðBÞ ¼ max kukH1l ; sup lðzÞjuðzÞj ln : 2 z2B 1 juðzÞj
However, in [16], we used the following slightly different norm on the Bloch space (i.e., when a ¼ 1)
kf k0Ba :¼ jf ð0Þj þ supð1 jzj2 Þa jrf ðzÞj: z2B
Some estimates of the essential norm of the operator uC u : Ba ðBÞ ðor Ba0 ðBÞÞ ! H1 l ðBÞ when a P 1, can be found in [14]. A natural question is to calculate the following operator norm
kuC u kBa ðBÞðor
ð1Þ
; Ba0 ðBÞÞ!H1 l ðBÞ
when a – 1: It turns out that the slight change of the definition of the norm k k0Ba on the a-Bloch space enables us to calculate (1) in an elegant way, which seems quite difficult for the case when the space is equipped with the norm k k0Ba . In order to prove the main result of this note we need an auxiliary result which could be folklore. Related estimates (up to the constant factor) are known, see, for example, Lemma 2.2 in [11]. Lemma 1. Let f 2 Ba ðBÞ; a – 1. Then the following inequality holds
! ba ðf Þ 1 jf ðzÞj 6 jf ð0Þj þ 1 : a 1 ð1 jzjÞa1
ð2Þ
Proof. Since a – 1, we have
Z jf ðzÞ f ð0Þj ¼
1
0
! Z 1 Z 1 d jzjdt ba ðf Þ 1 1 ðf ðtzÞÞdt ¼ hrf ðtzÞ; zidt 6 ba ðf Þ a ¼ dt a 1 ð1 jzjÞa1 0 0 ð1 jzjtÞ
from which the lemma easily follows. h 2. Norm of the operator uC u : Ba ðor Ba0 Þ ! H ‘ l Now we are in a position to formulate and prove the main result of this note. Theorem 1. Assume u 2 HðBÞ, u is a holomorphic self-map of B, a – 1, Then
(
kuC u kBa ðor Ba Þ!H1l ¼ max kukH1l ; sup 0
z2B
l is a weight, and uC u : Ba ðor Ba0 Þ ! H1 l is bounded. !)
lðzÞjuðzÞj 1 1 a1 ð1 juðzÞjÞa1
:
ð3Þ
Proof. Set f0 ðzÞ 1: It is easy to see that kf0 kBa ¼ 1 and f0 2 Ba0 . Hence we have
kuC u kBa !H1l ¼ kf0 kBa kuC u kBa !H1l P kuC u f0 kH1l ¼ kukH1l : 0
0
For each fixed w 2 B set
fw ðzÞ ¼
! 1 1 1 : a 1 ð1 hz; wiÞa1
Since fw ð0Þ ¼ 0 and
ð1 jzjÞa jrfw ðzÞj ¼
ð1 jzjÞa jwj ð1 jzjÞa ð1 jzjÞa a 6 a 6 min 1; a ; ð1 jwjjzjÞ ð1 jwjÞ j1 hz; wij
it follows that supw2B kfw kBa 6 1, and fw 2 Ba0 for each fixed w 2 B:
ð4Þ
S. Stevic´ / Applied Mathematics and Computation 215 (2009) 818–820
820
From this and the boundedness of uC u : Ba0 ! H1 l , for uðwÞ – 0 and every r 2 ð0; 1Þ we have
kuC u kBa !H1l P kuC u fruðwÞ=juðwÞj kH1l ¼ sup 0
z2B
lðzÞjuðzÞj 1 1 a 1 ja 1j ð1 rhuðzÞ; uðwÞi=juðwÞjÞ !
lðwÞjuðwÞj 1 P 1 : ða 1Þ ð1 rjuðwÞjÞa1
ð5Þ
If uðwÞ ¼ 0; then (5) obviously holds. Letting r ! 1 in (5), then taking the supremum over the unit ball in such obtained inequality, we get
!
kuC u kBa !H1l P sup 0
z2B
lðzÞjuðzÞj 1 1 : a1 ð1 juðzÞjÞa1
ð6Þ
From (4) and (6) it follows that
( kuC u kBa !H1l P max kukH1l ; sup 0
z2B
lðzÞjuðzÞj 1 1 a1 ð1 juðzÞjÞa1
!)
If f 2 Ba ; by Lemma 1 and the definition of the norm k kBa we get
kuC u f kH1l
ð7Þ
:
ba ðf Þ 1 ¼ sup lðzÞjuðzÞf ðuðzÞÞj 6 sup lðzÞjuðzÞj jf ð0Þj þ 1 a 1 ð1 juðzÞjÞa1 z2B z2B ( !) lðzÞjuðzÞj 1 1 6 kf kBa max kukH1l ; sup a1 z2B ð1 juðzÞjÞa1
!!
from which it follows that
(
!)
kuC u kBa !H1l 6 max kukH1l ; sup z2B
lðzÞjuðzÞj 1 1 a1 ð1 juðzÞjÞa1
:
ð8Þ
From (7) and (8) and the following obvious inequality
kuC u kBa !H1l 6 kuC u kBa !H1l 0
formula (3) follows, as claimed. h References [1] D. Clahane, S. Stevic´, Norm equivalence and composition operators between Bloch/Lipschitz spaces of the unit ball, J. Inequal. Appl. 2006 (Article ID 61018) (2006) 11p. [2] C.C. Cowen, B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, FL, 1995. [3] X. Fu, X. Zhu, Weighted composition operators on some weighted spaces in the unit ball, Abst. Appl. Anal. 2008 (Article ID 605807) (2008) 8p. [4] D. Gu, Weighted composition operators from generalized weighted Bergman spaces to weighted-type spaces, J. Inequal. Appl. 2008 (Article ID 619525) (2008) 14p. [5] S. Li, S. Stevic´, Weighted composition operators from Bergman-type spaces into Bloch spaces, Proc. Indian Acad. Sci. Math. Sci. 117 (3) (2007) 371–385. [6] S. Li, S. Stevic´, Weighted composition operators from a-Bloch space to H1 on the polydisk, Numer. Funct. Anal. Opt. 28 (7) (2007) 911–925. [7] S. Li, S. Stevic´, Weighted composition operators from H1 to the Bloch space on the polydisc, Abst. Appl. Anal. 2007 (Article ID 48478) (2007) 12p. [8] S. Li, S. Stevic´, Weighted composition operators between H1 and a-Bloch spaces in the unit ball, Taiwanese J. Math. 12 (2008) 1625–1639. [9] A. Montes-Rodriguez, Weighted composition operators on weighted Banach spaces of analytic functions, J. Lond. Math. Soc. 61 (3) (2000) 872–884. [10] S. Ohno, Weighted composition operators between H1 and the Bloch space, Taiwanese J. Math. 5 (2001) 555–563. [11] S. Stevic´, On an integral operator on the unit ball in Cn , J. Inequal. Appl. 1 (2005) 81–88. [12] S. Stevic´, Composition operators between H1 and the a-Bloch spaces on the polydisc, Z. Anal. Anwendungen 25 (4) (2006) 457–466. [13] S. Stevic´, Weighted composition operators between mixed norm spaces and H1 a spaces in the unit ball, J. Inequal. Appl. 2007 (Article ID 28629) (2007) 9p. [14] S. Stevic´, Essential norms of weighted composition operators from the a-Bloch space to a weighted-type space on the unit ball, Abst. Appl. Anal. 2008 (Article ID 279691) (2008) 11p. [15] S. Stevic´, Norms of some operators from Bergman spaces to weighted and Bloch-type space, Util. Math. 76 (2008) 59–64. [16] S. Stevic´, Norm of weighted composition operators from Bloch space to H1 l on the unit ball, Ars. Combin. 88 (2008) 125–127. [17] S. Stevic´, Norm and essential norm of composition followed by differentiation from a-Bloch spaces to H1 l , Appl. Math. Comput. 207 (2009) 225–229. [18] S. Stevic´, Weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball, Appl. Math. Comput. 212 (2009) 499–504. [19] S.I. Ueki, Weighted composition operators on some function spaces of entire functions, Bull. Belg. Math. Soc. Simon Stevin, in press. [20] S.I. Ueki, L. Luo, Compact weighted composition operators and multiplication opera-tors between Hardy spaces, Abst. Appl. Anal. 2008 (Article ID 196498) (2008) 11p. [21] S.I. Ueki, L. Luo, Essential norms of weighted composition operators between weighted Bergman spaces of the ball, Acta Sci. Math. (Szeged) 74 (2008) 829–843. [22] E. Wolf, Weighted composition operators between weighted Bergman spaces and weighted Banach spaces of holomorphic functions, Rev. Mat. Comput. 21 (2) (2008) 475–480. [23] C. Xiong, Norm of composition operators on the Bloch space, Bull. Aust. Math. Soc. 70 (2004) 293–299. [24] W. Yang, Weighted composition operators from Bloch-type spaces to weighted-type spaces, Ars. Combin., in press.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Norm of weighted composition operators from a-Bloch spaces to weighted-type spaces Stevo Stevic´ Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia
a r t i c l e
i n f o
Keywords: Weighted composition operator a-Bloch space Weighted-type space Operator norm
a b s t r a c t We calculate in an elegant way operator norm of the weighted composition operator from the a-Bloch space, with a 2 ð0; 1Þ n f1g, to a weighted-type space on the unit ball. This result can be regarded as a complement to our recent result regarding the same problem for the case a ¼ 1: Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction and preliminaries Let Bn ¼ B be the open unit ball in the complex vector space Cn , B1 ¼ D the open unit disk in C, HðBÞ the class of all holomorphic functions on the unit ball and H1 ðXÞ the space of all bounded holomorphic functions on a set X. For an f 2 HðBÞ we denote by rf the gradient of the function f, that is
rf ¼
@f @f ;...; : @z1 @zn
The a-Bloch space Ba ¼ Ba ðBÞ, consists of all f 2 HðBÞ such that
ba ðf Þ ¼ supð1 jzjÞa jrf ðzÞj < 1: z2B
It is easy to see that the a-Bloch space with the norm
kf kBa ¼ jf ð0Þj þ ba ðf Þ becomes a Banach space. The little a-Bloch space Ba0 is a subspace of Ba consisting of all f such that
lim ð1 jzjÞa jrf ðzÞj ¼ 0:
jzj!1
1 The weighted-type space H1 l ¼ Hl ðBÞ consists of all f 2 HðBÞ such that
kf kH1l :¼ sup lðzÞjf ðzÞj < 1; z2B
where l is a weight, i.e., a positive continuous function on B. Assume u 2 HðBÞ and u is a holomorphic self-map of B: The weighted composition operator induced by u and u is defined on HðBÞ by
ðuC u f ÞðzÞ ¼ uðzÞf ðuðzÞÞ: E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.06.005
S. Stevic´ / Applied Mathematics and Computation 215 (2009) 818–820
819
A typical problem is to provide function theoretic characterizations when u and u induce bounded or compact weighted composition operators between two given spaces of holomorphic functions (see, for example, [2]). The boundedness and compactness of weighted composition operators between the Bloch space and H1 ðDÞ was studied in [10]. The case of the unit polydisk was studied in [6,7] (the case of composition operators, i.e., when uðzÞ 1, was treated in [12]). The same problems in the setting of the unit ball were investigated in [8]. For related results see also [1,3– 5,9,13,20,21,23,24] and the references therein. It is of some interest to calculate operator norm of weighted composition operators. For some recent related results, see, e.g., [2,15–19,22]. In [16], we calculated operator norm of the operator uC u : BðBÞðor B0 ðBÞÞ ! H1 l ðBÞ. We proved the following formula
1 1 þ juðzÞj kuC u kBðBÞðor B0 ðBÞÞ!H1l ðBÞ ¼ max kukH1l ; sup lðzÞjuðzÞj ln : 2 z2B 1 juðzÞj
However, in [16], we used the following slightly different norm on the Bloch space (i.e., when a ¼ 1)
kf k0Ba :¼ jf ð0Þj þ supð1 jzj2 Þa jrf ðzÞj: z2B
Some estimates of the essential norm of the operator uC u : Ba ðBÞ ðor Ba0 ðBÞÞ ! H1 l ðBÞ when a P 1, can be found in [14]. A natural question is to calculate the following operator norm
kuC u kBa ðBÞðor
ð1Þ
; Ba0 ðBÞÞ!H1 l ðBÞ
when a – 1: It turns out that the slight change of the definition of the norm k k0Ba on the a-Bloch space enables us to calculate (1) in an elegant way, which seems quite difficult for the case when the space is equipped with the norm k k0Ba . In order to prove the main result of this note we need an auxiliary result which could be folklore. Related estimates (up to the constant factor) are known, see, for example, Lemma 2.2 in [11]. Lemma 1. Let f 2 Ba ðBÞ; a – 1. Then the following inequality holds
! ba ðf Þ 1 jf ðzÞj 6 jf ð0Þj þ 1 : a 1 ð1 jzjÞa1
ð2Þ
Proof. Since a – 1, we have
Z jf ðzÞ f ð0Þj ¼
1
0
! Z 1 Z 1 d jzjdt ba ðf Þ 1 1 ðf ðtzÞÞdt ¼ hrf ðtzÞ; zidt 6 ba ðf Þ a ¼ dt a 1 ð1 jzjÞa1 0 0 ð1 jzjtÞ
from which the lemma easily follows. h 2. Norm of the operator uC u : Ba ðor Ba0 Þ ! H ‘ l Now we are in a position to formulate and prove the main result of this note. Theorem 1. Assume u 2 HðBÞ, u is a holomorphic self-map of B, a – 1, Then
(
kuC u kBa ðor Ba Þ!H1l ¼ max kukH1l ; sup 0
z2B
l is a weight, and uC u : Ba ðor Ba0 Þ ! H1 l is bounded. !)
lðzÞjuðzÞj 1 1 a1 ð1 juðzÞjÞa1
:
ð3Þ
Proof. Set f0 ðzÞ 1: It is easy to see that kf0 kBa ¼ 1 and f0 2 Ba0 . Hence we have
kuC u kBa !H1l ¼ kf0 kBa kuC u kBa !H1l P kuC u f0 kH1l ¼ kukH1l : 0
0
For each fixed w 2 B set
fw ðzÞ ¼
! 1 1 1 : a 1 ð1 hz; wiÞa1
Since fw ð0Þ ¼ 0 and
ð1 jzjÞa jrfw ðzÞj ¼
ð1 jzjÞa jwj ð1 jzjÞa ð1 jzjÞa a 6 a 6 min 1; a ; ð1 jwjjzjÞ ð1 jwjÞ j1 hz; wij
it follows that supw2B kfw kBa 6 1, and fw 2 Ba0 for each fixed w 2 B:
ð4Þ
S. Stevic´ / Applied Mathematics and Computation 215 (2009) 818–820
820
From this and the boundedness of uC u : Ba0 ! H1 l , for uðwÞ – 0 and every r 2 ð0; 1Þ we have
kuC u kBa !H1l P kuC u fruðwÞ=juðwÞj kH1l ¼ sup 0
z2B
lðzÞjuðzÞj 1 1 a 1 ja 1j ð1 rhuðzÞ; uðwÞi=juðwÞjÞ !
lðwÞjuðwÞj 1 P 1 : ða 1Þ ð1 rjuðwÞjÞa1
ð5Þ
If uðwÞ ¼ 0; then (5) obviously holds. Letting r ! 1 in (5), then taking the supremum over the unit ball in such obtained inequality, we get
!
kuC u kBa !H1l P sup 0
z2B
lðzÞjuðzÞj 1 1 : a1 ð1 juðzÞjÞa1
ð6Þ
From (4) and (6) it follows that
( kuC u kBa !H1l P max kukH1l ; sup 0
z2B
lðzÞjuðzÞj 1 1 a1 ð1 juðzÞjÞa1
!)
If f 2 Ba ; by Lemma 1 and the definition of the norm k kBa we get
kuC u f kH1l
ð7Þ
:
ba ðf Þ 1 ¼ sup lðzÞjuðzÞf ðuðzÞÞj 6 sup lðzÞjuðzÞj jf ð0Þj þ 1 a 1 ð1 juðzÞjÞa1 z2B z2B ( !) lðzÞjuðzÞj 1 1 6 kf kBa max kukH1l ; sup a1 z2B ð1 juðzÞjÞa1
!!
from which it follows that
(
!)
kuC u kBa !H1l 6 max kukH1l ; sup z2B
lðzÞjuðzÞj 1 1 a1 ð1 juðzÞjÞa1
:
ð8Þ
From (7) and (8) and the following obvious inequality
kuC u kBa !H1l 6 kuC u kBa !H1l 0
formula (3) follows, as claimed. h References [1] D. Clahane, S. Stevic´, Norm equivalence and composition operators between Bloch/Lipschitz spaces of the unit ball, J. Inequal. Appl. 2006 (Article ID 61018) (2006) 11p. [2] C.C. Cowen, B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, FL, 1995. [3] X. Fu, X. Zhu, Weighted composition operators on some weighted spaces in the unit ball, Abst. Appl. Anal. 2008 (Article ID 605807) (2008) 8p. [4] D. Gu, Weighted composition operators from generalized weighted Bergman spaces to weighted-type spaces, J. Inequal. Appl. 2008 (Article ID 619525) (2008) 14p. [5] S. Li, S. Stevic´, Weighted composition operators from Bergman-type spaces into Bloch spaces, Proc. Indian Acad. Sci. Math. Sci. 117 (3) (2007) 371–385. [6] S. Li, S. Stevic´, Weighted composition operators from a-Bloch space to H1 on the polydisk, Numer. Funct. Anal. Opt. 28 (7) (2007) 911–925. [7] S. Li, S. Stevic´, Weighted composition operators from H1 to the Bloch space on the polydisc, Abst. Appl. Anal. 2007 (Article ID 48478) (2007) 12p. [8] S. Li, S. Stevic´, Weighted composition operators between H1 and a-Bloch spaces in the unit ball, Taiwanese J. Math. 12 (2008) 1625–1639. [9] A. Montes-Rodriguez, Weighted composition operators on weighted Banach spaces of analytic functions, J. Lond. Math. Soc. 61 (3) (2000) 872–884. [10] S. Ohno, Weighted composition operators between H1 and the Bloch space, Taiwanese J. Math. 5 (2001) 555–563. [11] S. Stevic´, On an integral operator on the unit ball in Cn , J. Inequal. Appl. 1 (2005) 81–88. [12] S. Stevic´, Composition operators between H1 and the a-Bloch spaces on the polydisc, Z. Anal. Anwendungen 25 (4) (2006) 457–466. [13] S. Stevic´, Weighted composition operators between mixed norm spaces and H1 a spaces in the unit ball, J. Inequal. Appl. 2007 (Article ID 28629) (2007) 9p. [14] S. Stevic´, Essential norms of weighted composition operators from the a-Bloch space to a weighted-type space on the unit ball, Abst. Appl. Anal. 2008 (Article ID 279691) (2008) 11p. [15] S. Stevic´, Norms of some operators from Bergman spaces to weighted and Bloch-type space, Util. Math. 76 (2008) 59–64. [16] S. Stevic´, Norm of weighted composition operators from Bloch space to H1 l on the unit ball, Ars. Combin. 88 (2008) 125–127. [17] S. Stevic´, Norm and essential norm of composition followed by differentiation from a-Bloch spaces to H1 l , Appl. Math. Comput. 207 (2009) 225–229. [18] S. Stevic´, Weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball, Appl. Math. Comput. 212 (2009) 499–504. [19] S.I. Ueki, Weighted composition operators on some function spaces of entire functions, Bull. Belg. Math. Soc. Simon Stevin, in press. [20] S.I. Ueki, L. Luo, Compact weighted composition operators and multiplication opera-tors between Hardy spaces, Abst. Appl. Anal. 2008 (Article ID 196498) (2008) 11p. [21] S.I. Ueki, L. Luo, Essential norms of weighted composition operators between weighted Bergman spaces of the ball, Acta Sci. Math. (Szeged) 74 (2008) 829–843. [22] E. Wolf, Weighted composition operators between weighted Bergman spaces and weighted Banach spaces of holomorphic functions, Rev. Mat. Comput. 21 (2) (2008) 475–480. [23] C. Xiong, Norm of composition operators on the Bloch space, Bull. Aust. Math. Soc. 70 (2004) 293–299. [24] W. Yang, Weighted composition operators from Bloch-type spaces to weighted-type spaces, Ars. Combin., in press.