ABSTRACT Pietro Poggi-Corradini. The research of Professor Pietro Poggi-Corradini stretches across the fields of complex analysis and functional analysis. The objects of study are on the one hand the analytic maps defined on the unit disk that are bounded by one in modulus, and on the other hand, the composition operators that these maps induce on classical function spaces of the disk. The motivations and the questions come from the theory of operators, but the tools used are drawn from complex analysis (harmonic measure, extremal distance, hyperbolic metric, etc...). The results, at least in the case when the map is one-to-one and has the origin as an attracting fixed point, are twofold. While describing properties of the associated composition operator, such as the spectrum and the essential spectral radius, new properties of the analytic map, such as the dynamical behavior near the boundary of the unit disk, were discovered. The link between the map and the operator is given by the linearization map near the fixed point provided by iteration theory, i.e. the Koenigs map. Properties of the Koenigs maps which had not been considered before, such as their Hardy class, are determined and these in turn shed light on the inner workings of the subject. P. Poggi-Corradini intends to continue his study in three different directions. The first consists in dropping the requirement that the maps be one-to-one. A second topic of inquiry is to ask similar questions on the Bergman spaces instead of the Hardy spaces. Many fundamental aspects of the theory of Bergman spaces have received much attention lately. A third line of research deals with the general problem of describing the spectrum of composition operators. It is clear that the properties of the spectrum and the linearization theory both depend on the location of the Denjoy-Wolff point. The results mentioned above only dealt with the case when this point does not have modulus one and the multiplier there is non-vanishing. So ther e is space for generalizing the techniques used to new situations. Finally, this study might help answer a question of M. Heins about generalizations of the Denjoy-Wolff Theorem to arbitrary Riemann surfaces and some questions of J. Cima about Cauchy-Stieltjes integrals. On a wider level, since, in recent years, both composition operators and complex dynamics have been studied intensively in the context of several complex variables, it would be interesting to see if the interplay between spectral theory of composition operators and the dynamics of analytic maps at the boundary of the disk carries over to self-maps of the ball or the polydisk. More generally, the research of P. Poggi-Corradini tackles fundamental issues in a classical mathematical field known as function theory, which has many applications to engineering and the applied sciences (e.g. iterative methods, aerodynamics, control theory, etc...). Much of the classical theory is devoted to the study of a single function or transformation. The innovative point of view is to extract information about a single transformation by iterating it repeatedly and analyzing the behavior "in the limit" of the ensuing dynamical system.

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Joe W. Jenkins
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University of Virginia
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