Generalized Functionals in Gaussian Spaces - The Characterization Theorem Revisited

Pusz--Woronowicz's functional calculus revisited

This note is a complement to Pusz--Woronowicz's works on functional calculus for two positive forms from the viewpoint of operator theory. Based on an elementary, self-contained and purely Hilbert space operator explanation of their functional calculus, we show that any operator connection type operations (including any operator perspectives) are captured by their functional calculus.

重新审视正算子的 Lebesgue 分解

Lebesgue decomposition for positive operators revisited

本文利用 Pusz–Woronowicz 泛函演算思想重新梳理 Ando 的 Hilbert 空间正算子 Lebesgue 分解理论，并重建其核心基础。

On Kakutani's characterization of the closed linear sublattices of $C(X)$ -- Revisited

In his paper [Concrete representation of abstract $(M)$-spaces. (A characterization of the space of continuous functions.), Ann. of Math., $42 (2)$ ($1941$), $994$--$1024$.], S. Kakutani gave an interesting representation of the closed linear sublattices of the space of real-valued continuous functions on a compact Hausdorff space, which is determined by a set of algebraic relations. In this short note, we present a simple proof of this representation without using any profound lattice theory or functional analysis results, making this proof accessible even to undergraduate students.

The Baillon-Haddad Theorem Revisited

In 1977, Baillon and Haddad proved that if the gradient of a convex and continuously differentiable function is nonexpansive, then it is actually firmly nonexpansive. This result, which has become known as the Baillon-Haddad theorem, has found many applications in optimization and numerical functional analysis. In this note, we propose short alternative proofs of this result and strengthen its conclusion.

概率空间中的指数不等式再探

Exponential inequalities in probability spaces revisited

该研究重新探讨概率空间指数可积性相关结果，通过新颖的对偶论证，推导出Moser-Trudinger型不等式框架下的量化新结论，复原高斯测度不等式，并将方法拓展至整数泊松测度的离散场景。

广义随机过程再探

Generalized stochastic processes revisited

本文研究广义随机过程中线性随机泛函的可测版本及唯一性，构建新框架，证明核空间之外空间的正则性及新的唯一性结果，并给出例证。