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The Essential Norm of a Weighted Composition Operator from the Bloch Space to H∞

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The Essential Norm of a Weighted Composition Operator from the Bloch Space to H∞ Bull. Aust. Math. Soc. 79 (2009), 487–497 doi:10.1017/S0004972709000094 THE ESSENTIAL NORM OF A WEIGHTED COMPOSITION OPERATOR FROM THE BLOCH SPACE TO H1 TAKUYA HOSOKAWA (Received 9 September 2008) Abstract We express the operator norm of a weighted composition operator which acts from the Bloch space B to H 1 as the supremum of a quantity involving the weight function, the inducing self-map, and the hyperbolic distance. We also express the essential norm of a weighted composition operator from B to H 1 as the asymptotic upper bound of the same quantity. Moreover we study the estimate of the essential norm of a weighted composition operator from H 1 to itself. 2000 Mathematics subject classification: primary 47B33. Keywords and phrases: weighted composition operator, the space of bounded analytic functions, Bloch space. 1. Introduction Let H.D/ be the set of all analytic functions on the open unit disk D and S.D/ the set of all analytic self-maps of D. Every analytic self-map ' 2 S.D/ induces a composition operator C' V f 7! f ◦ ' and every analytic function u 2 H.D/ induces a multiplication operator Mu V f 7! u · f . Both C' and Mu are linear transformations from H.D/ to itself. The weighted composition operator uC' is the product of Mu and C', that is, uC' f D MuC' f D u · f ◦ '. 1 1 1 Let H D H .D/ be the set of all bounded analytic functions on D. H is a Banach algebra with the supremum norm k f k1 D sup j f .z/j: z2D The Bloch space B is the set of all f 2 H.D/ satisfying kj f kj D sup.1 − jzj2/j f 0.z/j < 1: z2D This work was supported by the Korean Research Foundation Grant funded by Korean Government (KRF- 2008-314-C00012). c 2009 Australian Mathematical Society 0004-9727/2009 $16.00 487 Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 27 Sep 2021 at 06:34:56, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972709000094 488 T. Hosokawa [2] Then kj · kj defines a Möbius invariant complete semi-norm and B is a Banach space 1 under the norm k f kB D j f .0/j C kj f kj. Note that kj f kj ≤ k f k1 for any f 2 H , 1 hence H ⊂ B. Let the little Bloch space Bo denote the subspace of B consisting of those functions f such that lim .1 − jzj2/ f 0.z/ D 0: jzj!1 The little Bloch space Bo is a closed subspace of B. In particular, Bo is the closure in B of the polynomials. Let w be a point in D and αw be the Möbius transformation of D defined by w − z α .z/ D : w 1 − wz For w; z in D, the pseudo-hyperbolic distance ρ.w; z/ between z and w is given by w − z ρ.w; z/ D jαw.z/j D ; 1 − wz and the hyperbolic metric β.w; z/ is given by 1 1 C ρ.w; z/ β.w; z/ D log : 2 1 − ρ.w; z/ For f 2 B, we have that Z 1 jwj dt j f .w/ − f . /j ≤ kj f kj : 0 2 2 0 1 − jwj t Hence we have the growth condition for the Bloch functions: j f .w/j ≤ j f .0/j C kj f kjβ.w; 0/: (1.1) We have also the exact expression of the induced distance on the Bloch space: sup j f .z/ − f .w/j D β.z; w/: (1.2) k f kB≤1 See [9] for more information on the Bloch space. Let X and Y be two Banach spaces and T be a linear operator from X to Y . Denote the operator norm of T by kT kX!Y . If Y D X, we write kT kX D kT kX!X . For a bounded linear operator T from X to Y , the essential norm kT ke;X!Y is defined to be the distance from T to the closed ideal of compact operators, that is, kT ke;X!Y D inffkT C K kX!Y V K is compact from X to Y g: Notice that T is compact from X to Y if and only if kT ke;X!Y D 0. We also write kT ke;X D kT ke;X!X . 1 Ohno characterized the boundedness and the compactness of uC' from B to H . Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 27 Sep 2021 at 06:34:56, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972709000094 [3] Essential norm of a weighted composition operator 489 THEOREM A (Ohno et al. [4, 6]). Let u be in H.D/ and ' be in S.D/. 1 (i) The weighted composition operator uC' V B ! H is bounded if and only if 2 1 u H and e sup ju.z/j log < 1: − j j z2D 1 '.z/ 1 1 (ii) Suppose that uC' V B ! H is bounded. Then uC' V B ! H is compact if and only if u 2 H 1 and e lim sup ju.z/j log D 0: j'.z/j!1 1 − j'.z/j In [5], Kwon also studied the composition operators from B to H 1 and computed 0 1 0 the operator norm of C' from B to H with the condition '.0/ D 0, where B is the subspace of B which consists of all Bloch functions f with f .0/ D 0. THEOREM B (Kwon [5]). For any ' 2 S.D/ such that '.0/ D 0, we have kC'kB0!H 1 D sup β.'.z/; 0/: z2D In this paper, we estimate the operator norm and the essential norm of uC' acting 1 between B and H . We give the explicit formula of the operator norm of uC' from B to H 1 in Section 2, and of the essential norm in Section 3, which are the generalization of Theorems A and B. 1 Theorem A indicates that the compactness of uC' from B to H implies the compactness from H 1 to itself. Especially, for the case of composition operators, 1 1 the compactness of C' from B to H is equivalent to the compactness from H to 1 itself (see Corollary 3.4). The exact formula of the essential norm of uC' from H to itself has not been obtained. Takagi, Takahashi and Ueki gave a partial solution of kuC'ke;H 1 . THEOREM C (Takagi et al. [7]). Let u be in H.D/ and ' be in S.D/. For r > 0, let Dr D fz 2 D V ju.z/j ≥ rg and γ D inffr > 0 V '.Dr / ⊂ Dg: (1.3) Then γ ≤ kuC'ke;H 1 ≤ 2γ: So, in Section 4, we deal with the estimate of kuC'ke;H 1 under a certain assumption. 1 2. The operator norm of uC' from B to H 1 In this section, we consider the operator norm of uC' from B to H . For z 2 D, we shall use the short-hand notation β.e z/ D maxf1; β.z; 0/g: Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 27 Sep 2021 at 06:34:56, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972709000094 490 T. Hosokawa [4] THEOREM 2.1. Let u be in H.D/ and ' be in S.D/. Then we have 1 1 kuC'kB!H D kuC'kBo!H D sup ju.z/jβ.'.e z//: z2D 1 1 PROOF. Since Bo ⊂ B, it is easy to see that kuC'kBo!H ≤ kuC'kB!H . For f 2 B, the growth condition (1.1) implies kuC' f k1 D sup ju.z/ f .'.z//j z2D ≤ supfju.z/j.j f .0/j C kj f kj β.'.z///g z2D ≤ k f kB sup ju.z/jβ.'.e z//: z2D Hence we obtain kuC'kB!H 1 ≤ sup ju.z/jβ.'.e z//: z2D Next we show that 1 kuC'kBo!H ≥ sup ju.z/jβ.'.e z//: z2D To prove this, we use two test functions. The first one is the constant function 1. Since 1 is a unit vector of Bo, we have 1 kuC'kBo!H ≥ kuC' 1k1 D kuk1: (2.1) The second one is defined as following. Let r 2 .0; 1/ and λ 2 D. We put ηλ D λ=jλj and 1 1 C r ηλz fr,λ.z/ D log : 2 1 − r ηλz Then fr,λ.0/ D 0 and r.1 − jzj2/ .1 − jzj2/j f 0 .z/j D : r,λ 2 2 2 j1 − r ηλ z j Hence fr,λ 2 Bo and k fr,λkB D r. For arbitrary w 2 D, put λ D '.w/. Then kuC' fr;'.w/k1 1 kuC'kBo!H ≥ k fr;'.w/kB 1 1 1 C r η '.z/ D sup u.z/ log '.w/ − r z2D 2 1 r η'.w/'.z/ 1 ≥ ju.w/j β.r'.w/; 0/: r Downloaded from https://www.cambridge.org/core.
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