Mathematical Sciences: Lifting Properties of Invariant Forms and Applications

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The primary theme of this mathematical research is that of analyzing integral transforms of classical type in several new settings. They include transforms generated by toeplitz kernels and their liftings to abstract domains. A central feature of this analysis is the so-called generalized Bochner theorem which can be viewed as a Fourier representation of generalized invariant kernels. The integral representation feature is linked with the general theory of Krein and Gelfand on eigenfunction expansions, while the lifting property is related to the Nagy- Foias dilation theory. While the Nagy-Foias theory provides a powerful approach to the spectral theory of dissipative operators, clarifying the notions of scattering and characteristic functions, and includes interpolation theory, it is only remotely connected to the theories of Bochner and Krein. The purpose of this work is to explore the deeper connections between the different aspects of the Bochner theorem to understand the relations between the above-mentioned theories. In particular, work will be done in determining whether the Nagy-Foias commutant lifting is related to bound mean oscillation and prediction theory. In addition to direct application to complex interpolation and moment theory, this work is expected to have applications to scattering structures and control.