Shift Operators on Harmonic Hilbert Function Spaces on Real Unit Ball

Explore shift operators on harmonic function spaces in this 50-minute semi-plenary talk by H. Turgay Kaptanoğlu from Bilkent University. Delve into the definition of shift operators using zonal harmonics and partial derivatives, and examine their fundamental properties. Discover how these operators relate to multiplications by coordinate variables and projections on harmonic subspaces, leading to a new identity for zonal harmonics. Investigate large families of reproducing kernel Hilbert spaces of harmonic functions on the unit ball of R^n and analyze the action of shift operators on them. Learn about a dilation result for commuting row contractions of harmonic type and its implications for the maximality of certain norms. Understand the connection to von Neumann inequality for harmonic polynomials and the significance of the $\breve{\mathcal G}$ space as a natural harmonic counterpart to the Drury-Arveson space. This talk presents joint work with Daniel Alpay of Chapman University.