The evolution and applications of operator theory: From classical problems to scientific machine learning

This paper presents a conceptual and historical overview of operator theory as a unifying framework in mathematics and physics. From its roots in differential equations and boundary value problems, we trace the development of integral operators, spectral theory, and Hilbert space formalism, culminating in the axiomatization of unbounded and self-adjoint operators. We highlight the foundational role of operator algebras in quantum theory and noncommutative geometry, and explore modern applications in quantum information and data science. Emphasizing the spectral perspective, we show how operator theory connects analysis, geometry, and dynamics across classical and contemporary domains.