Beurling and Model subspaces invariant under a universal operator

Abstract

We classify the Beurling and model subspaces of the Hardy space $H^2(\mathbb{D})$ that are invariant under the composition operators $C_{{\varphi }_a}$, where ${\varphi }_a(z)=az+1-a$ and $0<a<1$. These operators have universal translates in the sense of Rota and provide a function-theoretic approach to the Hilbert space Invariant Subspace Problem. For each fixed $a$, a Beurling subspace $\Theta H^2$ is invariant precisely when \[ Θ(z)=λB(z)\exp\left(-c\,\frac{1+z}{1-z}\right), \qquad |λ|=1,\quad c\ge0, \] where $B$ is a possibly constant Blaschke product satisfying $m_B(w)\le m_B({\varphi }_a(w))$ for every $w\in \mathbb{D}$, with $m_B$ denoting zero multiplicity. For every affine self-map $\varphi (z)=\alpha z+\beta$ with $0<|\alpha |<1$, we show that the nonzero invariant model spaces are exactly the spaces of polynomials of bounded degree. We also recover the corresponding Beurling and model subspace classifications for the classical Cesàro operator. Finally, we characterize when a cyclic invariant subspace contains a nonzero function analytic across $1$ through its intersections with model spaces whose boundary spectra omit $1$, and exhibit an infinite-dimensional cyclic subspace containing an eigenvector but no such analytic function.

arXiv comment8 pages, revised and extended from the previous version. New content: converse of Theorem B and a new Section 6