Closed Range Composition Operators on the Bloch Space of Bounded Symmetric Domains
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Closed Range Composition Operators on the Bloch Space of Bounded Symmetric Domains Hidetaka Hamada Abstract. Let BX and BY be bounded symmetric domains realized as the unit balls of JB∗ -triples X and Y , respectively. In this paper, we generalize the Landau theorem to holomorphic mappings on BX using the Schwarz–Pick lemma for holomorphic mappings on BX . Next, we give a necessary condition for the composition operators Cϕ between the Bloch spaces on BX and BY to be bounded below by using a sampling set for the Bloch space, where ϕ is a holomorphic mapping from BX to BY . We also obtain other necessary conditions for the composition operators Cϕ between the Bloch spaces in the case BY is a complex Hilbert ball BH . We give a sufficient condition for the composition operators Cϕ between the Bloch spaces on BX and BY to be bounded below using a sampling set for the Bloch space. In the case dim X = dim Y < ∞, we also give another sufficient condition for the composition operators Cϕ between the Bloch spaces on BX and BY to be bounded below. Mathematics Subject Classification. Primary 32A18; Secondary 32M15, 47B38, 30H30. Keywords. Bloch space, bounded below, bounded symmetric domain, closed range, composition operator, Landau theorem.
1. Introduction The questions of boundedness and compactness of composition operators between the Bloch space B on the unit disc U in C has been studied by [18]. On the Euclidean unit ball in Cn , this problem has been investigated in [19,20]. On the other hand, about the boundedness below of composition operators on the Bloch space, Ghatage, Yan and Zheng [11] gave a necessary condition and proved that it is also sufficient under some restriction in the unit disc U in C. Chen [2] proved that the above necessary condition is still sufficient without any restriction, using the Landau theorem. He [2] also gave 0123456789().: V,-vol
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a necessary condition and a sufficient condition for the boundedness below of composition operators on the Bloch space of Bn . We remark that a sufficient condition was proved using the Landau theorem in several complex variables. Ghatage, Zheng and Zorboska [12] gave a necessary and sufficient condition for the composition operator on the Bloch space of U to be bounded below using a sampling set for the Bloch space. As a generalization to higher dimensions, Deng, Jiang and Ouyang [9] gave a necessary condition and a sufficient condition for the boundedness below of composition operators on the Bloch space of Bn using sampling sets for the Bloch space. In view of the Riemann mapping theorem, a homogeneous unit ball of a complex Banach space is a natural generalization of the open unit disc in C. Every bounded symmetric domain in a complex Banach space is biholomorphically equivalent to a homogeneous unit ball (see [15]). Bounded symmetric domains in a complex Banach space can be realized as the unit ball BX of a JB∗ -triple X. These arguments give the motivation for studying on the unit balls of JB∗ -triples. Recently, Chu, Hamada, Honda a
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