Universal Toeplitz operators on the Hardy space over the polydisk

TL;DR

This paper characterizes certain universal Toeplitz operators on the Hardy space over the polydisk, linking their properties to the invariant subspace problem and providing conditions for universality in higher dimensions.

Contribution

It introduces a characterization of analytic Toeplitz operators on the polydisk whose adjoints are universal, extending the understanding of invariant subspaces in multivariable Hardy spaces.

Findings

Operators with inner function symbols are universal on the polydisk.

Polynomial symbols with zeros inside the polydisk but zero-free on the torus are universal.

Results differ from the classical case when n=1.

Abstract

The Invariant Subspace Problem (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota), the ISP may be solved by describing the invariant subspaces of these operators alone. We characterize all anaytic Toeplitz operators $T_\phi$ on the Hardy space $H^2(\mathbb{D}^n)$ over the polydisk $\mathbb{D}^n$ for $n>1$ whose adjoints satisfy the Caradus criterion for universality, that is, when $T_\phi^*$ is surjective and has infinite dimensional kernel. In particular if $\phi$ in a non-constant inner function on $\mathbb{D}^n$, or a polynomial in the ring $\mathbb{C}[z_1,\ldots,z_n]$ that has zeros in $\mathbb{D}^n$ but is zero-free on $\mathbb{T}^n$, then $T_\phi^*$ is universal for $H^2(\mathbb{D}^n)$. The analogs of these results for $n=1$ are not true.