Mathematical Sciences: Abstract Spectral Decompositions and Transference Methods in Modern Analysis

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Professor Berkson conducts research on the interface of harmonic analysis, ergodic theory, and operator theory. All three are important subdisciplines of Modern Analysis. Harmonic analysis, roughly speaking, embodies the interrelationships of mathematical analysis and group actions. Historically, harmonic analysis has its origins in Fourier series and partial differential equations. The related field of ergodic theory originated in statistical mechanics. Its thrust is in the understanding and exploitation of the concept of recurrence. Both ergodic theory and harmonic analysis, as throughout all of mathematical analysis, utilize methods and operations that are susceptible to broad abstract principles. Modern operator theory, which arose in the development of the mathematical foundation of quantum physics, seeks to identify and explicate these abstract principles so as to unify and expand the frontiers of science. The subject matter of this proposal is operator theory in this conceptual sense, with emphasis on applications to ergodic theory and harmonic analysis. Professor Berkson proposes to continue his development of an abstract operator theory that encompasses weakened forms of orthogonality (such as Fourier inversion and triangular truncation in noncommutative analysis). The approach is based upon the notions of trigonometrically well-bounded operator and spectral family of projections which are used in the general settings of analysis to replace, respectively, the unitary operators and projection-valued measures central to Hilbert space operator theory and its physical applications. In prior collaborative research, Professor Berkson has developed these notions, along with extensive applications to commutative and noncommutative analysis. The current investigations will expand and sharpen the abstract operator theory along these lines and its companion vector-valued transference methods, while extending its scope to such topics as representations of locally compact abelian groups, Fourier multiplier transference, ergodic theory, and generalized analyticity.