Abstract
Abstract In this note, we study defect operators in the case of holomorphic functions of the unit ball of ℂn. These operators are built from weighted Bergman kernel with a holomorphic vector. We obtain a description of sub-Hilbert spaces and we give a sufficient condition so that theses spaces are the same.
Highlights
In this note, we study defect operators in the case of holomorphic functions of the unit ball of Cn
These operators are built from weighted Bergman kernel with a holomorphic vector
We obtain a description of sub-Hilbert spaces and we give a su cient condition so that theses spaces are the same
Summary
The space (I − H) (Aβ) is a sub-weighted Bergman space of Aβ with the reproducing kernel given by (I − H)Bβ(z, w) = Kβ(z, w). Burbea [3], for spaces of holomorphic functions on domains of Cn and by S. Saitoh for abstract kernels of Hilbert spaces of functions [10]. Chu [4] for weighted Bergman spaces. The following proposition gives a description of the sub-Hilbert spaces : Theorem 1. For −n ≤ β < − , Bβ(z, w) is the reproducing kernel of the diagonal weighted Besov space Bβ : A holomorphic function f belongs to Bβ if and only if (I + N)m f is in A m+β, where. Equality of weighted sub-Bergman Hilbert spaces means equality as set
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