泛函分析：2025–2026 年度综述

An annual survey of the math.FA arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 120 papers and organised into 44 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• Spectral Theory and Operator Semigroups (56.2%) — Focuses on spectral properties, stability, and dynamics of linear and nonlinear operators, semigroups, and matrices in Banach and Hilbert spaces.
• Banach Space Geometry and Linear Operators (24.6%) — Investigates geometric properties, boundedness, and structural characteristics of linear operators and function spaces within Banach and Hilbert frameworks.
• Topological and Metric Aspects of Functional Analysis (4.2%) — Examines convergence, compactness, and continuity properties in metric spaces, Lie algebras, and function spaces, often involving local or uniform structures.
• General Theorems and Operator Algebra Frameworks (3.7%) — Develops and extends classical theorems and general frameworks in operator theory and operator algebras, focusing on theoretical foundations and characterizations.
• Existence and Uniqueness in Functional Equations (2.9%) — Addresses existence, uniqueness, and representation problems for solutions to functional equations, moment problems, and nonlinear systems.
• Sharp Norm Inequalities and Optimal Transport (2.3%) — Derives sharp bounds, optimal constants, and stability estimates for norm inequalities and optimal transport problems in various metric and functional settings.
• Semigroups, Kernels, and Fixed Point Theory (2.1%) — Studies spectral theory of semigroups, reproducing kernels, and fixed point properties in locally convex spaces and operator models.
• Fourier Analysis and Measure Theory Applications (1.8%) — Explores Fourier transforms, phase retrieval, and measure-theoretic properties including uncertainty principles and probability measure transformations.
• Operator Algebras and Von Neumann Structures (1.2%) — Analyzes structural properties of von Neumann algebras, operator systems, and related algebraic frameworks including trace and entropy functionals.
• Algebraic Structures in Functional Calculus (0.8%) — Investigates algebraic and tensor structures within functional calculus, convolution algebras, and operator models in specific functional settings.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.