Noncommutative analysis, Multivariable spectral theory for operators in Hilbert space, Probability, and Unitary Representations

TL;DR

This paper explores the development of noncommutative analysis and multivariable spectral theory within Hilbert spaces, emphasizing applications across mathematics, physics, and signal processing, and aims to make the subject more accessible and flexible for diverse learners.

Contribution

It introduces a flexible, application-driven approach to functional analysis, integrating noncommutative analysis and spectral theory with minimal prerequisites and practical motivation.

Findings

Enhanced understanding of spectral theory in noncommutative settings

Application-driven approach improves accessibility for students from diverse backgrounds

Framework supports interdisciplinary applications in physics, probability, and signal processing

Abstract

Over the decades, Functional Analysis has been enriched and inspired on account of demands from neighboring fields, within mathematics, harmonic analysis (wavelets and signal processing), numerical analysis (finite element methods, discretization), PDEs (diffusion equations, scattering theory), representation theory; iterated function systems (fractals, Julia sets, chaotic dynamical systems), ergodic theory, operator algebras, and many more. And neighboring areas, probability/statistics (for example stochastic processes, Ito and Malliavin calculus), physics (representation of Lie groups, quantum field theory), and spectral theory for Schr\"odinger operators. We have strived for a more accessible book, and yet aimed squarely at applications; -- we have been serious about motivation: Rather than beginning with the four big theorems in Functional Analysis, our point of departure is an initial choice of topics from applications. And we have aimed for flexibility of use; acknowledging that students and instructors will invariably have a host of diverse goals in teaching beginning analysis courses. And students come to the course with a varied background. Indeed, over the years we found that students have come to the Functional Analysis sequence from other and different areas of math, and even from other departments; and so we have presented the material in a way that minimizes the need for prerequisites. We also found that well motivated students are easily able to fill in what is needed from measure theory, or from a facility with the four big theorems of Functional Analysis. And we found that the approach "learn-by-using" has a comparative advantage.