Commuting tuple of multiplication operators homogeneous under the unitary group

• Published in: Journal of the London Mathematical Society 2024-04, Vol.109 (4), p.n/a
• Main Authors: Ghara, Soumitra, Kumar, Surjit, Misra, Gadadhar, Pramanick, Paramita
• Format: Article
• Language: English
• ISSN: 0024-6107; 1469-7750
• DOI: 10.1112/jlms.12890

Let U(d)$\mathcal {U}(d)$ be the group of d×d$d\times d$ unitary matrices. We find conditions to ensure that a U(d)$\mathcal {U}(d)$‐homogeneous d$d$‐tuple T$\bm{T}$ is unitarily equivalent to multiplication by the coordinate functions on some reproducing kernel Hilbert space HK(Bd,Cn)⊆Hol(Bd,Cn)$\mathcal {H}_K(\mathbb {B}_d, \mathbb {C}^n) \subseteq \mbox{\rm Hol}(\mathbb {B}_d, \mathbb {C}^n)$, n=dim∩j=1dkerTj∗$n= \dim \cap _{j=1}^d \ker T^*_{j}$. We describe this class of U(d)$\mathcal {U}(d)$‐homogeneous operators, equivalently, nonnegative kernels K$K$ quasi‐invariant under the action of U(d)$\mathcal {U}(d)$. We classify quasi‐invariant kernels K$K$ transforming under U(d)$\mathcal {U}(d)$ with two specific choice of multipliers. A crucial ingredient of the proof is that the group SU(d)$SU(d)$ has exactly two inequivalent irreducible unitary representations of dimension d$d$ and none in dimensions 2,…,d−1$2, \ldots , d-1$, d⩾3$d\geqslant 3$. We obtain explicit criterion for boundedness, reducibility, and mutual unitary equivalence among these operators.