Isometries of some classical function spaces among the composition operators
Isometries of some classical function spaces among the composition operators
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DOI：
10.1090/conm/393/07375
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
María J. Martín;D. Vukotić
中科院分区：
文献类型：
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作者：
María J. Martín;D. Vukotić
我们给出了所有可能的组合算子的特征的简单而统一的证明，这些组合算子是圆盘的一般哈代空间或一般加权伯格曼空间的等距。我们对具有单价符号的复合算子之间的解析贝索夫空间（包含狄利克雷空间）的等距进行同样的操作。简介 在本文中，dm(θ) = (2π)−1dθ 将表示单位圆 T 上的归一化弧长度量。我们假设读者熟悉圆盘的标准哈代空间 H 的定义（例如，参见 [D]）。我们将 dA 写为单位圆盘 D 上的归一化勒贝格面积度量：dA(re) = π−1rdrdθ。加权伯格曼空间 Aw 是圆盘中解析的所有 L (D, w dA) 函数的空间，其中 w 是径向权重函数：w(z) = w(|z|)，非负且关于 dA 可积。当 1 ≤ p < ∞ 时，每个 H 都是 Banach 空间，当权重 w “合理”时，Aw 也是 Banach 空间（只要点评估有界；粗略地说，在单位圆附近，w 不应该“经常”为零）。当 w == 1 时，得到未加权的伯格曼空间 A（有关这些空间的理论，请参见[DS]）；标准加权空间 Aα 对应于 w(z) = (α + 1)(1− |z|2)α, −1 < α < Infini 的情况。给定单位圆盘 D 中的解析函数 φ，使得 φ(D) ⊂ D，根据 Littlewood 的从属定理，具有由 Cφf(z) = f(φ(z)) 定义的符号 φ 的复合算子 Cφ 始终有界于任何 H 或 Aα 空间。专着 [S1] 和 [CM] 是此类空间上的合成算子理论的标准来源。 2000年数学学科分类。初级 47B33；中学 30H05。
We give a simple and unified proof of the characterizations of all possible composition operators that are isometries of either a general Hardy space or a general weighted Bergman spaces of the disk. We do the same for the isometries of analytic Besov spaces (containing the Dirichlet space) among the composition operators with univalent symbols. Introduction Throughout this note, dm(θ) = (2π)−1dθ will denote the normalized arc length measure on the unit circle T. We assume that the reader is familiar with the definition of the standard Hardy spaces H of the disk (see [D], for example). We write dA for the normalized Lebesgue area measure on the unit disk D: dA(re) = π−1rdrdθ. The weighted Bergman space Aw is the space of all L (D, w dA) functions analytic in the disk, where w is a radial weight function: w(z) = w(|z|), non-negative and integrable with respect to dA. Every H is a Banach space when 1 ≤ p < ∞, and so is Aw when the weight w is “reasonable” (whenever the point evaluations are bounded; roughly speaking, w should not be zero “too often” near the unit circle). The unweighted Bergman space A is obtained when w ≡ 1 (see [DS] for the theory of these spaces); the standard weighted space Aα corresponds to the case w(z) = (α + 1)(1− |z|2)α, −1 < α < ∞. Given an analytic function φ in the unit disk D such that φ(D) ⊂ D, the composition operator Cφ with symbol φ defined by Cφf(z) = f(φ(z)) is always bounded on any H or Aα space, in view of Littlewood’s Subordination Theorem. The monographs [S1] and [CM] are standard sources for the theory of composition operators on such spaces. 2000 Mathematics Subject Classification. Primary 47B33; Secondary 30H05.