文摘
We continue the study of spherically balanced Hilbert spaces initiated in the first part of this paper. Recall that the complex Hilbert space \(H^2(\beta )\) of formal power series in the variables \(z_1, \ldots , z_m\) 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 the variable \(t\) such that $$\begin{aligned} \Vert f\Vert ^2_{H^2(\beta )} = \int _{\partial {\mathbb {B}}}\Vert {f_z}\Vert ^2_{H^2(\gamma )}~d\mu (z)~(f \in H^2(\beta )), \end{aligned}$$where \(f_z(t)=f(t z)\) is a formal power series in the variable \(t\). In the first half of this paper, we discuss operator theory in spherically balanced Hilbert spaces. The first main result in this part describes quasi-similarity orbit of multiplication tuple \(M_z\) on a spherically balanced space \(H^2(\beta ).\) We also observe that all spherical contractive multi-shifts on spherically balanced spaces admit the classical von Neumann’s inequality. In the second half, we introduce and study a class of Hilbert spaces, to be referred to as \({\mathcal {G}}\)-balanced Hilbert spaces, where \({\mathcal {G}}={\mathcal {U}}(r_1) \times {\mathcal {U}}(r_2) \times \cdots \times {\mathcal {U}}(r_k)\) is a subgroup of \({\mathcal {U}}(m)\) with \(r_1 + \cdots + r_k=m.\) In the case in which \({\mathcal {G}}={\mathcal {U}}(m),\) \({\mathcal {G}}\)-balanced spaces are precisely spherically balanced Hilbert spaces. Keywords Multi-shifts Slice representation Spherical isometry Quasi-similarity Von Neumann’s inequality Mathematics Subject Classification Primary 47A13 46E20 47B37 Secondary 47B20 Communicated by Heinrich Begehr.