2008-2009
UF MATHEMATICS COLLOQUIUM

Abstract. A classical theorem of Herglotz represents each holomorphic function with positive real part in the unit disk as the Herglotz transform of a positive measure on the unit circle. (An equivalent theorem of F. Riesz represents positive harmonic functions as Poisson integrals of positive measures.) We will discuss the Herglotz representation from the point of view of operator theory and functional analysis. In particular we will introduce an operator-valued version of the Herglotz kernel and apply it to the study of functions of positive real part in the unit ball of Cⁿ, which in general are much less tractable.

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