广义双圆幂等元与 $C^1[0,1]$ 上等距反射的结构
Structure of Generalized bi-circular idempotents and isometric reflections on $C^1[0,1]$
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中文总结 AI 辅助
本文研究 $C^1[0,1]$ 上满足正交分解条件的幂等映射对，刻画了其线性组合成为等距的充要条件，并分类了广义双圆与双圆幂等元族，同时给出了等距反射的完整结构。
中文摘要 AI 辅助
设 $C^1[0,1]$ 为所有在单位区间 $[0,1]$ 上连续可微的复值函数 $f$ 构成的复线性空间，其范数为 $\\|f\\|_{\sigma} = |f(0)| + \\|f'\\|_\infty$。设 $P_1, P_2: C^1[0,1] \rightarrow C^1[0,1]$ 为互不相同、非零的幂等映射（不一定是线性的），满足 $P_1P_2 = P_2P_1 = 0$ 且 $P_1+P_2 = I$，其中 $I$ 表示恒等算子。本文证明了：对于某些互不相同的单位模复数 $\lambda_1, \lambda_2$，$\lambda_1P_1 + \lambda_2P_2$ 是 $C^1[0,1]$ 上的等距，当且仅当要么 $\lambda_1 + \lambda_2 = 0$，要么 $\lambda_1P_1 + \lambda_2P_2$ 对所有这样的复数 $\lambda_1, \lambda_2$ 都是等距。在前一种情形下，集合 $\{P_1, P_2\}$ 被称为广义双圆幂等元族；在后一种情形下，被称为双圆幂等元族。本文还刻画了 $C^1[0,1]$ 上等距反射（即满足 $T^2 = I$ 的等距 $T$）的结构，并讨论了其与上述幂等映射类的关系。
英文摘要
Let $C^1[0,1]$ be the complex linear space of all continuously differentiable complex-valued functions $f$ on the unit interval $[0,1]$ with respect to the norm $ \|f\|_σ = |f(0)| +\|f'\|_\infty$. Let $P_1, P_2: C^1[0,1] \rightarrow C^1[0,1]$ be distinct, nonzero idempotent maps, which are not necessarily linear, such that $P_1P_2 = P_2P_1 = 0$ and $P_1+P_2 = I$, where $I$ denotes the identity operator. It is proved that $λ_1P_1 + λ_2P_2$ is an isometry on $C^1[0,1]$, for some distinct unit modulus complex numbers $λ_1, λ_2$, if and only if either $λ_1 + λ_2 = 0$, or $λ_1P_1 + λ_2P_2$ is an isometry for all such complex numbers $λ_1, λ_2$. In the former case, the collection $\{P_1, P_2\}$ is called a family of generalized bi-circular idempotents; in the latter case, it is called a family of bi-circular idempotents. The structure of isometric reflection on $C^1[0,1]$, that is, an isometry $T$ such that $T^2 = I$, is characterized, and its relationship with the above class of idempotents maps is also discussed.
发表机构
- Indian Institute of Information Technology Allahabad（印度信息技术学院阿拉哈巴德分校）
机构由 AI 辅助整理，请以论文原文为准。
