算子的分数幂：通过δ-正则化对数表示的刻画
Fractional Powers of Operators: Characterization via $δ$-Regularized Logarithmic Representation
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中文总结 AI 辅助
本文基于巴拿赫空间C⁰-半群理论，通过δ-正则化对数表示构造算子分数幂，突破经典方法的约束，证明更一般C⁰-半群生成元的分数幂良定，应用潜力广泛。
中文摘要 AI 辅助
对于算子的分数幂，V. Balakrishnan的经典公式固有地要求生成元具有正性或特定的扇形条件（即解析半群的生成）。为克服这些限制，本文基于巴拿赫空间中抽象发展方程的C⁰-半群理论框架，提出一种通过无穷小生成元D的δ-正则化对数表示来构造算子分数幂Dᵏ的新方法。该方法通过对有界算子采用代数δ-正则化，绕过了传统的几何约束。具体而言，通过将复解析对数表示应用于由半群预解式导出的一族有界算子，我们从数学上证明，在可逆性这一最小代数假设下，更一般C⁰-半群生成元的分数幂可唯一且严格地良定，且与正则化参数δ的选择无关。本研究建立的理论框架具有广泛的应用潜力，包括非解析半群和无穷小生成元显式依赖于时间变量的非自治系统的正则性估计。
英文摘要
For the fractional powers of operators, the classical formulation by V. Balakrishnan inherently requires positivity or specific sectorial conditions of the generator (i.e., the generation of analytic semigroups). To overcome these limitations, this paper presents a novel approach to constructing fractional powers of operators $D^k$ through the $δ$-regularized logarithmic representation of infinitesimal generators $D$, based on the framework of $C^0$-semigroup theory for abstract evolution equations in Banach spaces. The proposed method bypasses the conventional geometric constraints by employing an algebraic $δ$-regularization for bounded operators. Specifically, by applying a complex-analytic logarithmic representation to a family of bounded operators derived from the resolvent of the semigroup, we mathematically prove that, under the minimal algebraic assumption of invertibility, the fractional powers for generators of more general $C^0$-semigroups are uniquely and rigorously well-defined, independent of the choice of the regularization parameter $δ$. The theoretical framework established in this study offers extensive potential for applications, including regularity estimates for non-analytic semigroups and non-autonomous systems where the infinitesimal generators depend explicitly on the time variable.
