Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 348208, 15 pages doi:10.1155/2008/348208 Research Article Riemann-Stieltjes Operators between Vector-Valued Weighted Bloch Spaces Maofa Wang School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China Correspondence should be addressed to Maofa Wang, mfwang.math@whu.edu.cn Received 6 June 2008; Revised 1 August 2008; Accepted 11 September 2008 Recommended by Jozsef Szabados Let X be a Banach space, we study Riemann-Stieltjes operators between X-valuedweightedBloch spaces. Some necessary and sufficient conditions for these operators induced by holomorphic functions to be weakly compact and weakly conditionally compact are given by certain growth properties of the inducing symbols and some structural properties of the abstract Banach space, which extend some previous results. Copyright q 2008 Maofa Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and statement of the main results Let D  {z ∈ C : |z| < 1} be the open unit disk in the complex plane C. Denote by HD the space of all holomorphic functions on D. For g ∈ HD, the Riemann-Stieltjes operator T g is defined on HD by  T g f  z  z 0 fζdgζ  1 0 ftzzg  tzdt, z ∈ D. 1.1 The Riemann-Stieltjes operator T g can be viewed as a generalization of the well-known Ces ` aro operator defined by  Cf  z 1 z  z 0 fζ 1 − ζ dζ  ∞  n0  1 n  1 n  i0 a i  z n ,z∈ D 1.2 for fz  ∞ n0 a n z n ∈ HD. Pommerenke 1 initiated the study of Riemann-Stieltjes operators on Hardy space H 2 , where he proved that T g is bounded on H 2 if and only if g is in BMOA, the space of 2 Journal of Inequalities and Applications holomorphic functions on D with bounded mean oscillation. T his result later was extended to other Hardy spaces H p ,1 <p<∞ see 2. Similar questions on weighted Bergman spaces were considered by Aleman and Siskakis in 3: T g is bounded on Bergman space A 2 if and only if g is in Bloch space. Henceforward, many papers have been published which discuss the action of Riemann-Stieltjes operators on distinct spaces of holomorphic functions, including Hardy spaces, weighted Bergman spaces, Dirichlet spaces, BMOA, VMOA, Bloch spaces, and so on; see, for example, 4–9 and the related references therein. Among the prominent results we mention the characterization of Riemann-Stieltjes operators on Bloch space in terms of the growth properties of the inducing symbols 8, where Yoneda proved that T g is bounded on the Bloch space B if and only if sup z∈D  1 −|z| 2   log 1 1 −|z| 2    g  z   < ∞; 1.3 T g is compact on the Bloch space B if and only if lim |z|→1 −  1 −|z| 2   log 1 1 −|z| 2    g  z    0, 1.4 where the Bloch space B : {f ∈ HD :sup z∈D 1 −|z| 2 |f  z| < ∞}. Recently, several authors have published papers to extend this result from different angles. Some papers discussed a higher dimensional version Riemann-Stieltjes operator of 1.1 to the unit ball B n of C n replacing g  z by the radial derivative Rg of g. For example, Hu 10 gave the characterizations of bounded and compact Riemann-Stieltjes operators on the Bloch space of B n ,Xiao11 further studied the Riemann-Stieltjes operators on weighted Bloch and Bergman spaces of the unit ball, Zhang 9 studied the boundedness and compactness of Riemann- Stieltjes operators on Dirichlet-type spaces and Bloch-type spaces of B n , on general Bloch- type spaces, the Riemann-Stieltjes operators were studied in 5, 12. From the main result of 9see also in 12, 13, we know that T g : B α →B β is bounded if and only if g ∈B β for 0 < α<1; sup z∈D 1 −|z| 2  β |g  z|log2/1 −|z| 2  < ∞ for α  1; and sup z∈D 1 −|z| 2  β1−α |g  z| < ∞ for α>1, where α, β > 0andB α : {f ∈ HD :sup z∈D 1 −|z| 2  α |f  z| < ∞}. One can further refer to 10 –12, 14, 15 for more study of Riemann-Stieltjes operators on Hardy spaces, Bergman spaces, and Bloch spaces of t he unit ball B n . It is worth remarking that all the above spaces which T g targets are not beyond a spectrum of the scalar-valued holomorphic function spaces. The purpose of this paper is to initiate the study of Riemann-Stieltjes operators on spaces of vector-valued holomorphic functions. Let X be any complex Banach space and α>0, the vector-valued weighted Bloch space B α X consists of all X-valued holomorphic functions f : D → X such that sup z∈D  1 −|z| 2  α   f  z   X < ∞. 1.5 The little weighted Bloch space B α 0 X is the subspace of B α X consisting of the holomorphic functions f : D → X for which lim |z|→1 − 1 −|z| 2  α f  z X  0. For f ∈B α X, define f B α X    f0   X  sup z∈D 1 −|z| 2  α   f  z   X . 1.6 Maofa Wang 3 With this norm, both B α X and B α 0 X are Banach spaces. These classes of vector-valued spaces have been studied quite extensively; see, for instance, 16, 17. For simplification, we often write B α and B α 0 instead of B α C and B α 0 C, respectively. For more information on the scalar-valued Bloch spaces, one can refer to 18, 19. When α  1, we often omit the α from B α X. Clearly, f ∈B α X if and only if x ∗ ◦f·x ∗ f· ∈B α for all x ∗ ∈ X ∗ , the dual space of X. Moreover, f B α X ≈ sup x ∗  X ∗ ≤1 x ∗ ◦ f B α . Here and in the sequel, we write a  b or b  a for any nonnegative quantities a and b if a is dominated by b times some inessential positive constant, and write a ≈ b for a  b  a. Since x ∗  T g f  zx ∗   z 0 fζdgζ    z 0 x ∗  fζ  dgζT g  x ∗ f  z1.7 for any x ∗ ∈ X ∗ and f ∈B α X, T g is bounded between B α X and B β X if and only if it is bounded between the corresponding scalar-valued spaces B α and B β . In addition, in Section 3, we will see that when the Banach space X is infinite-dimensional, T g is never compact between B α X and B β X except for the trivial case that g is a constant function. In this paper, we will study some small property of Riemann-Stieltjes operators between X-valued Bloch spaces. The main goal is to generalize some characterizations of compact Riemann-Stieltjes operators on scalar-valued Bloch spaces to the vector-valued case. Our main result is for the weak compactness of T g . Theorem 1.1. Let α, β > 0, X be a complex Banach space and g : D → C a nonconstant holomorphic function. Then the following hold. 1 For 0 <α<1,T g : B α X →B β X (resp., T g : B α 0 X →B β 0 X) is weakly compact if and only if X is reflexive and sup z∈D  1 −|z| 2  β   g  z   < ∞  resp., lim |z|→1 −  1 −|z| 2  β   g  z    0  . 1.8 2 For α  1,T g : B α X →B β X (or T g : B α 0 X →B β 0 X) is weakly compact if and only if X is reflexive and lim |z|→1 −  1 −|z| 2  β  ln 2 1 −|z| 2    g  z    0. 1.9 3 For α>1,T g : B α X →B β X (or T g : B α 0 X →B β 0 X) is weakly compact if and only if X is reflexive and lim |z|→1 −  1 −|z| 2  β−α1   g  z    0. 1.10 Theorem 1.1 illustrates that T g is weakly compact between B α X and B β X if and only if X is reflexive and T g is compact between the corresponding scalar-valued spaces. It also illustrates that the weak compactness of T g depends on α with α>1,butthisisnot 4 Journal of Inequalities and Applications the case when α ∈ 0, 1, however, for the case α  1, the condition needs an additional logarithmic term. The rest of the paper is organized as follows. We give some lemmas in Section 2, which are essentially needed for our proof of the main result. The proof of Theorem 1.1 and the reason why we do not consider the compactness of T g are given in Section 3. Finally, we briefly consider the weakly conditional compactness of T g between B α X and B β X and obtain some counterpart of our main result. In the sequel, we often use the same letter C, depending only on the allowed parameters, to denote various positive constants which may change at each occurrence. 2. Preliminaries First, we need the following growth estimate of Bloch functions. Lemma 2.1. For α>0 and any complex Banach space X,iff ∈B α X,then 1 fz X  f B α X for any z ∈ D and 0 <α<1; 2 fz X  ln2/1 −|z| 2 f BX for any z ∈ D and α  1; 3 fz X  1/1 −|z| 2  α−1 f B α X for any z ∈ D and α>1. Proof. Since for any x ∗ ∈ X ∗ ,    x ∗ ◦ f   z   ≤   x ∗ ◦ f   B α  1 −|z| 2  α ,z∈ D, 2.1 so   x ∗ ◦ fz − x ∗ ◦ f0   ≤  1 0    x ∗ ◦ f   zt   |z|dt ≤   x ∗ ◦ f   B α  1 0 |z|  1 − t 2 |z| 2  α dt. 2.2 Taking the supremum over x ∗ in the unit ball of X ∗ and estimating the last integral will give the desired results. For little Bloch spaces, we have the following improved behavior of f near the boundary ∂D. Lemma 2.2. Let X be a Banach space. 1 If f ∈B 0 X,then lim |z|→1 −   fz   X ln  2/  1 −|z| 2   0. 2.3 2 If f ∈B α 0 X with α>1,then lim |z|→1 −  1 −|z| 2  α−1   fz   X  0. 2.4 Maofa Wang 5 Proof. Since f ∈B α 0 X, then lim |z|→1 − 1 −|z| 2  α f  z X  0, that is, lim |z|→1 −  1 −|z| 2  α sup x ∗  X ∗ ≤1   x ∗ ◦ f  z    0. 2.5 So for any ε>0, there is r 0 ∈ 0, 1 such that  1 −|z| 2  α   x ∗ ◦ f  z   <ε for any r 0 < |z| < 1,   x ∗   X ∗ ≤ 1. 2.6 Then for any r 0 < |z| < 1, x ∗ ∈ X ∗ with x ∗  X ∗ ≤ 1, we have   x ∗ ◦ fz − x ∗ ◦ f0   ≤  r 0 /|z| 0    x ∗ ◦ f   zt   |z|dt   1 r 0 /|z|    x ∗ ◦ f   zt   |z|dt  f B α X  r 0 /|z| 0 |z|dt  1 −|zt| 2  α  ε  1 r 0 /|z| |z|dt  1 −|zt| 2  α  f B α X  ε  1 r 0 /|z| |z|dt  1 −|zt| 2  α , 2.7 the fact x ∗ ◦ f  zx ∗ ◦ f  z is used in the second inequality above. Since  1 r 0 /|z| |z|dt  1 −|zt| 2  α  ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ln 2 1 −|z| 2 ,α 1, 1  1 −|z| 2  α−1 ,α>1, 2.8 taking the supremum over all x ∗ with x ∗  X ∗ ≤ 1 and a variance of 2.7 will complete the proof. The following lemma is based on the well-known properties of the de la Vall ´ ee-Poussin summability kernel, which is used to approximate T g in the operator norm by suitable weakly compact operators sequence. Lemma 2.3. For α>0 and any complex Banach space X, there are linear operators {V n } on B α X satisfying the following properties. 1 V n ≤3 for any n ≥ 1. In addition, V n B α 0 X ⊂B α 0 X. 2 For every r ∈ 0, 1, lim n →∞ sup ||f|| B α X ≤1 sup |z|≤r f − V n fz X  0. 3 If X is reflexive (resp., does not contain a copy of l 1 ), then V n is weakly compact (resp., weakly conditionally compact) on B α X for all n ≥ 1. 6 Journal of Inequalities and Applications Proof. We first define the operators  V n by setting  V n fz n  k0 a k z k  2n−1  kn1 2n − k n a k z k 2.9 for any holomorphic function f : D → X with the Taylor expansion fz  ∞ k0 a k z k .Note that  V n fz  2K 2n−1 − K n−1  ∗fz 1 2π  2π 0  2K 2n−1 θ − K n−1 θ  f  ze −iθ  dθ, 2.10 where K n θ  n k−n 1 −|k|/n  1e ikθ denotes the Fej ´ er kernel, which is a summability kernel, that is, 1/2π  2π 0 K n θdθ  1 refer to 20. Then we have    V n f   H ∞ X ≤ 3f H ∞ X , 2.11 where · H ∞ X denotes the norm on the X-valued Hardy space H ∞ X given by f H ∞ X  sup z∈D fz X . For any ε>0andr ∈ 0, 1, there exists n 0 > 0 such that r n ≤ ε/4forn>n 0 . Given f ∈ H ∞ X, we write f −  V n f  z n g, then g H ∞ X  sup z∈D   gz   X  sup z∈∂D   gz   X  sup z∈∂D   z n gz   X  sup z∈D   z n gz   X  sup z∈D   fz −  V n fz   X    f −  V n f   H ∞ X , 2.12 the second equality above is due to the subharmonicity of gz X .So   fz −  V n fz   X  |z| n   gz   X ≤ r n g H ∞ X ≤ ε 4   f −  V n f   H ∞ X ≤ εf H ∞ X 2.13 Maofa Wang 7 for n>n 0 and all |z| <r, the last inequality comes from 2.11. Now we define the desired operators {V n } via  V n as follows: V n fzf0  z 0  V n f  ζdζ 2.14 for any holomorphic function f : D → X.ClearV n f is holomorphic and actually V n fz n1  k0 a k z k  2n  kn2 2n  1 − k n a k z k 2.15 for any holomorphic function fz  ∞ k0 a k z k . Since sup |z|r    V n f   z   X  sup |z|r    V n f  z   X  sup |z|1    V n f  rz   X     V n f  r   H ∞ X ≤ 3   f  r   H ∞ X  3sup |z|r   f  z   X , 2.16 where f r ·fr· for any 0 <r<1. Hence   V n f   B α X ≤ 3f B α X ,f∈B α X, 2.17 by the definition of the norm · B α X and V n f0f0. In addition, it is clear that V n B α 0 X ⊂B α 0 X,sinceV n f is always a polynomial by 2.15. This completes the proof of part 1. Since fzf0  z 0 f  ζdζ, V n fzf0  z 0  V n f  ζdζ, 2.18 so   fz − V n fz   X       1 0  f  zt −  V n f  zt  zdt     X . 2.19 8 Journal of Inequalities and Applications Hence for any |z|≤r<1,   fz − V n fz   X ≤  1 0 sup |z|≤r    f  zt −  V n f  zt    X dt   1 0 sup |z|≤ √ r   f  √ r zt −  V n f  √ r zt   X dt ≤  1 0 ε1 − r α   f  √ r   H ∞ X dt  ε1 − r α   f  √ r   H ∞ X ≤ εf B α X 2.20 for large enough n, the third inequality to the last is followed by applying 2.13 to the function f  √ r and the constant ε1 − r α . This completes the proof of part 2. Finally, for any n, define S n f   a 0 ,a 1 , ,a 2n  2.21 for any holomorphic function f : D → X with Taylor expansion fz  ∞ k0 a k z k ,anddefine  T n χ  z n1  k0 a k z k  2n  kn2 2n  1 − k n a k z k 2.22 for any χ a 0 ,a 1 , ,a 2n  ∈   2n 0 X l 2 . I t is clear that S n : B α X →   2n 0 X l 2 and T n :   2n 0 X l 2 →B α X are well defined and bounded. Moreover, V n  T n S n by 2.15, that is, V n has a factorization through   2n 0 X l 2 . It follows from Alaoglu’s theorem 21 and Rosenthal’s l 1 -criterion 22 that V n is weakly compact resp., weakly conditionally compact if X is reflexive resp., does not contain a copy of l 1 . The proof is complete. 3. Proof of the main results Before proving Theorem 1.1, we first recall that a bounded linear operator T : E → F from the Banach space E to the Banach space F is compact resp., weakly compact if every bounded sequence {f n }⊂E has a subsequence {f n k } such that {Tf n k } is norm convergent resp., weakly convergent. A useful characterization for a bounded linear operators to be weakly compact is the Gantmacher’s theorem 21: T is weakly compact if and only if T ∗∗ E ∗∗  ⊂ F, where T ∗∗ is the second adjoint of T,andE ∗∗ is the second dual of E. Notice that if g is a nonconstant holomorphic function such that T g : B α X →B β X or T g : B α 0 X →B β 0 X is compact, then for any bounded sequence {x n } in X and f n z ≡ x n , {f n } is a bounded sequence of B α 0 X since f n  B α X  x n  X , and then there exists a subsequence {f n k } by the definition of compact operators such that {T g f n k } is norm convergent in B β X. On the other hand, F n z : T g f n z  z 0 x n g  ζdζ  x n  gz − g0  . 3.1 Maofa Wang 9 So {F n k } is norm convergent in B β X. It follows from Lemma 2.1 that F n k converges uniformly on any compact subset of D, especially it is pointwise convergent. That is, for any bounded sequence {x n }⊂X, there is a subsequence {x n k } such that it is norm convergence in X since g is nonconstant, so X must be finite-dimensional Banach space by Bolzano- Weierstrass theorem 21. Namely, for any infinite-dimensional Banach space X, T g is never compact between X-valued weighted Bloch spaces except for the trivial case that gwhich is a constant function. From here on, we always assume that X is an infinite-dimensional Banach space, similar analysis as above shows that if the Riemann-Stieltjes operator T g is weakly compact from B α X to B β Xor from B α 0 X to B β 0 X, then for f n z ≡ x n , a bounded sequence in X, there exists a subsequence {f n k } such that {T g f n k } is weakly convergent. Without loss of generality, we may assume that {T g f n k } converges weakly to 0. Fix any z ∈ D,letδ z be the point evaluation function at z,thatis,δ z ffz,f ∈B β X. Then for any x ∗ ∈ X ∗ ,the functional x ∗ ◦ δ z fx ∗  fz  ,f∈B β X3.2 satisfies   x ∗ ◦ δ z f   ≤   x ∗   X ∗   fz   X  C z   x ∗   X ∗ f B β X ,f∈B β X3.3 for some constant C z > 0byLemma 2.1.Thatis,x ∗ ◦ δ z ∈ B β X ∗ ,sox ∗ ◦ δ z T g f n k  → 0as k →∞,thatis, x ∗  x n k  gz − g0  −→ 0, as k −→ ∞, 3.4 for any x ∗ ∈ X ∗ and z ∈ D. Since g is nonconstant, so x ∗ x n k  → 0ask →∞for any x ∗ ∈ X ∗ . That is, for any bounded sequence {x n }⊂X, there is a subsequence {x n k } such that it is weakly convergent, then X must be reflexive refer to 23. Namely, the reflexivity of X is a necessary condition for the weak compactness of T g between X-valued Bloch spaces. Under this assumption, Theorem 1.1 states that T g : B α X →B β Xor T g : B α 0 X →B β 0 X is weakly compact if and only if the corresponding scalar-valued operator T g is compact by the main result in 9. We are now going to complete the proof of Theorem 1.1. Proof of Theorem 1.1. We first assume that X is reflexive and define the operator V n as in Lemma 2.3,thatis, V n fz n1  k0 a k z k  2n  kn2 2n  1 − k n a k z k 3.5 for any holomorphic function fz  ∞ k0 a k z k .FromLemma 2.3, we know that the operators {V n } are all weakly compact on B α X and uniform bounded. So it suffices to prove that the norm T g − T g V n →0asn →∞under the conditions 1.8, 1.9,and1.10, 10 Journal of Inequalities and Applications respectively, because the weakly compact operators form a closed operator ideal. For any f ∈B α X,weknowf − V n f ∈B α X and  1 −|z| 2  β    T g − T g V n  f   z   X   1 −|z| 2  β   g  z     fz − V n fz   X : Az. 3.6 For α  1, if lim |z|→1 − 1 −|z| 2  β ln2/1 −|z| 2 |g  z|  0, then for arbitrary ε>0, there is r ∈ 0, 1 such that 1 −|z| 2  β ln2/1 −|z| 2 |g  z| <εfor |z| >r.So Az  1 −|z| 2  β   g  z      I − V n  fz   X   1 −|z| 2  β  ln 2 1 −|z| 2    g  z      I − V n  fz   X ln  2/  1 −|z| 2   εf B α X 3.7 for |z| >rby Lemmas 2.12 and 2.31. And for |z|≤r, Az  1 −|z| 2  β   g  z      I − V n  fz   X     I − V n  fz   X  εf BX 3.8 for large enough n by Lemma 2.32. Hence T g − T g V n  <εfor nsufficiently large. This completes the sufficiency for the case α  1 since at this time we again have T g B 0 X ⊂ B β 0 X. Similarly, for α>1, if lim |z|→1 − 1 −|z| 2  β−α1 |g  z|  0, then for arbitrary ε>0, there is r ∈ 0, 1 such that 1 −|z| 2  β−α1 |g  z| <εfor |z| >r.So Az  1−|z| 2  β   g  z      I−V n  fz   X   1 −|z| 2  β−α1   g  z      I − V n  f   B α X εf B α X 3.9 for |z| >rby Lemmas 2.13 and 2.31. Hence T g −T g V n  <εfor n sufficiently large by 3.8. This completes the sufficiency for the case α>1 since again T g B α 0 X ⊂B β 0 X. For α ∈ 0, 1, the method above does not work. We complete the proof by the definition of weak compactness of T g . Since g satisfies 1.8, it is obvious that T g : B α X →B β Xresp., T g : B α 0 X →B β 0 X is bounded. For any bounded sequence {f n }⊂ B α X, we have   f n z   X    f n   B α X  1,z∈ D, 3.10 by Lemma 2.1. 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Inequalities and Applications Volume 2008, Article ID 348208, 15 pages doi:10.1155/2008/348208 Research Article Riemann-Stieltjes Operators between Vector-Valued Weighted Bloch Spaces Maofa Wang School of. Szabados Let X be a Banach space, we study Riemann-Stieltjes operators between X-valuedweightedBloch spaces. Some necessary and sufficient conditions for these operators induced by holomorphic functions. the characterizations of bounded and compact Riemann-Stieltjes operators on the Bloch space of B n ,Xiao11 further studied the Riemann-Stieltjes operators on weighted Bloch and Bergman spaces of the unit ball,