Journal of Operator Theory

Volume 72, Issue 2, Fall 2014 pp. 405-428.

Spherically balanced Hilbert spaces of formal power series in several variables. I

Authors: Sameer Chavan (1) and Surjit Kumar (2)
Author institution: (1) Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur-208016, India
(2) Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur-208016, India

Summary: Motivated by theory of spherical Cauchy dual tuples, we study the spherically balanced spaces, that is, Hilbert spaces $H^2(\beta)$ of formal power series in the variables $z_1, \ldots, z_m$ for which $\{\beta_n\}_{n \in \mathbb Z^m_+}$ satisfies \beqn \sum_{k=1}^m \frac{\beta^2_{n+ \varepsilon_i + \varepsilon_k}}{\beta^2_{n+\varepsilon_i}} = \sum_{k=1}^m \frac{\beta^2_{n+ \varepsilon_j + \varepsilon_k}}{\beta^2_{n+ \varepsilon_j}}\quad\mbox{for~all~}n \in \mathbb Z^m_+~\mbox{and~}i, j = 1, \ldots, m. \eeqn The main result in this paper states that $H^2(\beta)$ is spherically balanced if and only if there exist a Reinhardt measure $\mu$ supported on the unit sphere $\partial \mathbb B$ and a Hilbert space $H^2(\gamma)$ of formal power series in one variable such that $\begin{equation*} \|f\|^2_{H^2(\beta)} = \int\limits_{\partial \mathbb B}\|{f_z}\|^2_{H^2(\gamma)}\mathrm d\mu(z)\quad(f \in H^2(\beta)). \end{equation*}$

DOI: http://dx.doi.org/10.7900/jot.2013apr22.2000
Keywords: multi-shifts, slice representation, spherical isometry, cyclic vectors

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