Mathematical Sciences: RUI: Spaces of Holomorphic Functions and Their Operators

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9502983 Lotto This project involves work on spaces of holomorphic functions and operators on these spaces.. Particular problems in the proposal include the classification of the multipliers and interpolating sequences for de Branges-Rovnyak spaces, the boundedness of products of (possibly unbounded) Toeplitz and Hankel operators, the relationship of the noncyclic vectors of the backward shift to operator ranges, the classification of rigid functions in Hardy spaces and the behavior of such functions under composition, the use of disintegration of measures in the study of composition operators, the study of von Neumann's inequality for three or more commuting diagonalizable contractions, the perturbation of commuting operators to commuting diagonalizable operators and the cyclicity of the compression of analytic multiplication operators to an invariant subspace of the backward shift operator. Hilbert space operators are infinite generalizations of matrices. The infinite generalization of a vector is frequently a function and for this reason Hilbert space operators are frequently modeled as the operator of multiplication on a space of functions. This project involves the study of many basic problems of multiplication models for operators acting on a variant of Hilbert spaces called de Branges-Rovnyak spaces. The theme that connects all these problems is the interplay among operator theory, functional analysis, and complex function theory. The interplay provides a richer and deeper understanding of all three of these topics than one could hope for by studying them individually ***