Hardy Spaces and Toeplitz operators on the Pentablock

Abstract

The pentablock is a domain in $\mathbb C^3$ arising in the study of the $μ$-synthesis problem and is defined by \[ \mathcal P = \left\{ (a_{21},\operatorname{tr}(A),\det(A)): A=[a_{ij}]_{i,j=1}^2\in M_2(\mathbb C),\ \|A\|<1 \right\}. \] We introduce the Hardy space $H^2(\mathcal P)$ associated with the canonical measure $μ$ on the distinguished boundary $b\mathcal P$. We identify $H^2(\mathcal P)$ with a closed subspace of $L^2(b\mathcal P,μ)$ and show that it is a reproducing kernel Hilbert space. We establish a new description of $\mathcal P$ and its closure and obtain a uniqueness set. In particular, we prove that $\mathcal P$ is the interior of the polynomial convex hull of $Ψ(\overline{\mathbb D}^3)$, where \[ Ψ(ξ,λ_1,λ_2) = \left( \frac{ξ(λ_1-λ_2)}{2}, λ_1+λ_2, λ_1λ_2 \right) \] is the parametrization map for $\mathcal P$. We also show that $H^2(\mathcal P)$ embeds isometrically as a closed subspace of $H^2(\mathbb D^3)$. We then obtain a Brown-Halmos type characterization of Toeplitz operators on $H^2(\mathcal P)$ and a commutant lifting theorem for $\mathcal P$-isometries and their minimal $\mathcal P$-unitary extensions. As consequences, we determine the commutant of the coordinate multipliers, characterize the compact Toeplitz operators and show that the coordinate multipliers have no nontrivial joint reducing subspaces. Finally, we characterize $\mathcal P$-unitaries as images of commuting triples of unitaries under $Ψ$ and present a class of $\mathcal P$-contractions from triples having $\overline{\mathbb D}^3$ as a spectral set.