turkmath.org

In an attempt to identify the harmonic Drury-Arveson space, we introduce and investigate large families of reproducing kernel Hilbert spaces of harmonic functions the unit ball of $\mathbb R^n$. Using zonal harmonics, we define and develop basic properties of shift operators and their adjoints in the harmonic setting. We prove a dilation result for the shift operators on harmonic spaces that are row contractions. As a consequence, we show that the norm of one of our spaces Ğ is maximal among those spaces on which the shift operator is a row contraction. We also show the maximality of the operator norm of the shift on Ğ among contractive Hilbert norms on harmonic polynomials. We then describe the progress towards a von Neumann inequality for harmonic polynomials and a tuple of commuting operators on harmonic spaces that are row contractions and belong to a certain class. This presentation is based on joint works with Ebrahim Samei and Varvara Shepelska of University of Saskatchewan, Canada.