Operator theory on generalized Hartogs triangles

Abstract

We consider the family $\mathcal P$ of $n$-tuples $P$ consisting of polynomials $P_1, \ldots, P_n$ with nonnegative coefficients which satisfy $\partial_i P_j(0) = \delta_{i, j},$ $i, j=1, \ldots, n.$ With any such $P,$ we associate a Reinhardt domain $\triangle^{\!n}_{_P}$ that we will call the generalized Hartogs triangle. We are particularly interested in the choices $P_a = (P_{1, a}, \ldots, P_{n, a}),$ $a \geq 0,$ where $P_{j, a}(z) = z_j + a \prod_{k=1}^n z_k,~ j=1, \ldots, n.$ The generalized Hartogs triangle associated with $P_a$ is given by \begin{equation} \triangle^{\!n}_a = \Big\{z \in \mathbb C \times \mathbb C^{n-1}_* : |z_j|^2 < |z_{j+1}|^2(1-a|z_1|^2), ~j=1, \ldots, n-1, |z_n|^2 + a|z_1|^2 < 1\Big\}. \end{equation} The domain $\triangle^{\!n}_{_P},$ $n \geq 2$ is never polynomially convex. However, $\triangle^{\!n}_{_P}$ is always holomorphically convex. With any $P \in \mathcal P$ and $m \in \mathbb N^n,$ we associate a positive semi-definite kernel $\mathscr K_{_{P, m}}$ on $\triangle^{\!n}_{_P}.$ This combined with the Moore's theorem yields a reproducing kernel Hilbert space $\mathscr H^2_m(\triangle^{\!n}_{_P})$ of holomorphic functions on $\triangle^{\!n}_{_P}.$ We study the space $\mathscr H^2_m(\triangle^{\!n}_{_P})$ and the multiplication $n$-tuple $\mathscr M_z$ acting on $\mathscr H^2_m(\triangle^{\!n}_{_P}).$ It turns out that $\mathscr M_z$ is never rationally cyclic. Although the dimension of the joint kernel of $\mathscr M^*_z-\lambda$ is constant of value $1$ for every $\lambda \in \triangle^{\!n}_{_P}$, it has jump discontinuity at the serious singularity $0$ of the boundary of $\triangle^{\!n}_{_P}$ with value equal to $\infty.$ We capitalize on the notion of joint subnormality to define a Hardy space on $\triangle^{\!n}_{_0}.$ This in turn gives an analog of the von Neumann's inequality for $\triangle^{\!n}_{_0}.$