Nell Paiz Jacobsson: Clark measures of bivariate inner functions

Abstract.

To each inner function on the unit disc, one can associate a corresponding family of Clark measures, which in turn can be linked to a family of unitary operators. Hence Clark measures form a link between inner functions, singular measures and operator theory. While Clark theory in one variable has been thoroughly studied and well-developed following D. N. Clark’s 1972 paper, it is only recently that progress has been made in extending this theory to the multivariate setting. Our goal is to provide an overview of recent research as well as investigate Clark measures for some new examples of bivariate inner functions on the unit bidisc. In particular, we characterize the Clark measures for certain kinds of multiplicative embeddings in inner functions as well as products of inner functions.