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Composition operators between weighted spaces of holomorphic functions on Banach spaces. (English) Zbl 1064.47022

Let \(X\) be a complex Banach space and \(B_X\) its open unit ball. We denote by \(H(B_X)\) the space of all holomorphic functions \(f:B_X \to {\mathbb C}\). \(H_v(B_X)\) is the space of all functions \(f\) in \(H(B_X)\) such that \(| | f| | _v = \sup_{x \in B_X} v(x) | f(x)| < \infty\), where \(v: B_X \to (0, \infty )\) is a continuous bounded mapping called a weight. In this paper, the authors study the behavior of composition operators \(C_\phi: H_v(B_Y) \to H_w(B_X)\), \(C_\phi f:= f \circ \phi\) (\(X\), \(Y\) Banach spaces; \(\phi:B_X \to B_Y\) holomorphic; \(v\), \(w\) weights). Some characterizations of the boundedness of \(C_\phi\) are given. To get one of them, an additional condition is necessary, as is shown with an example.
About the compactness, several necessary and sufficient conditions are obtained. From them, the finite-dimensional case is characterized, but as several nice examples show, it is not possible to get, in general, the converse of the mentioned conditions in the infinite-dimensional setting. Thus, in this last case a new phenomenon appears.
The authors also obtain conditions on a weight \(v\) which make sure the continuity of \(C_\phi: H_v(B_H) \to H_v(B_X)\), where now \(H\) is a Hilbert space.
Many of the results in this paper improve other ones in J. Bonet, P. Domanshi, M. Lindström and J. Taskinen [J. Aust. Math. Soc., Ser. A 64, No. 1, 101–118 (1998; Zbl 0912.47014)].

MSC:

47B33 Linear composition operators
46E15 Banach spaces of continuous, differentiable or analytic functions
46G20 Infinite-dimensional holomorphy

Citations:

Zbl 0912.47014
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