发表机构
University of Chicago; Columbia University（芝加哥大学; 哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对无界算子建立了三种自然形式的奇异值分解，填补了该领域的空白，并展示了其在多个数学和物理领域中的应用与新颖见解。
AI 中文摘要
奇异值分解已针对矩阵、Hilbert-Schmidt算子、迹类算子、紧算子和有界算子建立，但令人惊讶的是，尚未针对无界算子建立。不幸的是，应用数学中大多数有趣的算子都是无界的，因为任何涉及某种导数形式的算子——梯度、外导数、拉普拉斯算子、Fourier变换及其他导数变换、Hamiltonian等——都可能是无界的。在本文中，我们填补了这一最后缺失的空白，通过三种自然形式建立了无界算子奇异值分解的存在性：乘法算子形式、直接积分形式和算子值测度形式。我们证明它继承了有限维奇异值分解的经典性质，包括逼近结果、与基本子空间的关系以及Moore-Penrose逆。这一发现为数学、物理学、统计学和金融学中众多著名无界算子的奇异值分解打开了大门——包括欧几里得空间和流形上的梯度、Petrov-Galerkin方法、有限差分算子、Hilbert-Schmidt算子、Hilbert复形、超对称量子力学、Sturm-Liouville理论、非参数密度估计和Black-Scholes方程。所得的分解揭示了许多新颖的见解，其中包括：超对称量子力学中的玻色子和费米子态作为广义阶梯算子的左、右奇异向量出现；Riesz变换作为欧几里得梯度的左奇异算子出现；Hodge分解直接由外导数的奇异值分解得出。
英文摘要
The singular value decomposition has been established for matrices, Hilbert--Schmidt operators, trace-class operator, compact operators, and bounded operators, but surprisingly not for unbounded operators. Unfortunately, most interesting operators in applied math are unbounded, as any operators involving some form of derivatives --- gradient, exterior derivatives, Laplacians, Fourier and other transforms of derivatives, Hamiltonians, etc. --- are likely unbounded. In this article, we fill in this last missing piece by establishing the existence of singular value decompositions for unbounded operators in three natural forms: multiplication-operator, direct-integral, and operator-valued-measure. We show it inherits classical properties of finite-dimensional singular value decomposition including approximation results, relationships with fundamental subspaces, and the Moore--Penrose inverse. This discovery opens the door to the singular value decompositions of a myriad of well-known unbounded operators in mathematics, physics, statistics, and finnance --- gradients on Euclidean spaces and manifolds, Petrov--Galerkin method, finite-difference operators, Hilbert--Schmidt operators, Hilbert complexes, supersymmetric quantum mechanics, Sturm--Liouville theory, nonparametric density estimation, and the Black--Scholes equation. The resulting decompositions reveal a number of novel insights, among many others: bosonic and fermionic states in supersymmetric quantum mechanics arise as left and right singular vectors of generalized ladder operators; the Riesz transform appears as the left singular operator of the Euclidean gradient; and the Hodge decomposition follows directly from the singular value decompositions of the exterior derivatives.
