Isometries of a Bergman-Privalov-Type Space on the Unit Ball
نویسندگان
چکیده
We introduce a new space ANlog,α B consisting of all holomorphic functions on the unit ball B ⊂ C such that ‖f‖ANlog,α : ∫ B φe ln 1 |f z | dVα z < ∞, where α > −1, dVα z cα,n 1 − |z| dV z dV z is the normalized Lebesgue volume measure on B, and cα,n is a normalization constant, that is, Vα B 1 , and φe t t ln e t for t ∈ 0,∞ . Some basic properties of this space are presented. Among other results we proved that ANlog,α B with the metric d f, g ‖f − g‖ANlog,α is an F-algebra with respect to pointwise addition and multiplication. We also prove that every linear isometry T of ANlog,α B into itself has the form Tf c f ◦ ψ for some c ∈ C such that |c| 1 and some ψ which is a holomorphic self-map of B satisfying a measure-preserving property with respect to the measure dVα. As a consequence of this result we obtain a complete characterization of all linear bijective isometries of ANlog,α B .
منابع مشابه
Multiplicative Isometries on F-Algebras of Holomorphic Functions
and Applied Analysis 3 Now we recall definitions and some properties of the Smirnov class, the Privalov class, the Bergman-Privalov class, and the Zygmund F-algebra on Bn or D. The space of all holomorphic functions on X Bn or D is denoted by H X . For each 0 < p ≤ ∞, the Hardy space is denoted by H X with the norm ‖ · ‖p. 2.1. Smirnov Class N∗ X Let X ∈ {Bn, Dn}. The Nevanlinna class N X on X ...
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