Shift operators on harmonic Hilbert function spaces on real balls and von Neumann inequality (Q831109)

The goal of the paper under review is to establish a multivariate von Neumann inequality for harmonic polynomials on \(\mathbb{R}^n\). The authors first construct shift operators \(({S}_1,\ldots,{S}_n)\) on the space of harmonic functions. This is a nontrivial task since harmonicity is not preserved under multiplication. Second, a harmonic version of the Drury-Arveson space is defined as the reproducing kernel Hilbert space \(\breve{\mathcal{G}}\) of functions on the unit ball in \(\mathbb{R}^n\) having kernel given by \[ \breve{G}(x,y) = \sum_{m=0}^{\infty}\frac{1}{A_m}Z_m(x,y), \] where each \(Z_m(x,y)\) is a zonal harmonic polynomial of degree \(m\) and \(A_m\) is the coefficient of \((x\cdot y)^m\) in the series expansion of \(Z_m(x,y)\). In order to prove their version of von Neumann inequality, the authors restrict themselves to tuples \((T_1,\ldots, T_n)\) of commuting operators that satisfy \(T_1T_1 + \cdots + T_nT_n = 0\). Such tuples are said to be of \textit{harmonic type}. The restriction \((\breve{S}_1,\ldots,\breve{S}_n)\) of the harmonic shift \((S_1,\ldots,S_n)\) on \(\breve{\mathcal{G}}\) is proved to be a row contraction of harmonic type. Among several results in the paper, the following theorem offers a multivariate von Neumann inequality for harmonic polynomials: Let \((T_1,\ldots, T_n)\) be a harmonic-type commuting row contraction on a Hilbert space. If \(u\) is a harmonic polynomial in \(n\) variables, then \[ \|u(T_1,\ldots,T_n)\|\leq \|u(\breve{S}_1,\ldots,\breve{S}_n)\|. \]