A representation theorem for cyclic analytic two-isometries

Abstract:

A bounded linear operator $T$ on a complex separable Hilbert space $\mathcal {H}$ is called a $2$-isometry if ${T^{\ast 2}}{T^2} - 2{T^{ \ast }}T + I = 0$. We say that $T$ is analytic if ${ \cap _{n> 0}} {T^n} \mathcal {H}= (0)$. In this paper we show that every cyclic analytic $2$-isometry can be represented as multiplication by $z$ on a Dirichlet-type space $D(\mu )$. Here $\mu$ denotes a finite positive Borel measure on the unit circle. For two measures $\mu$ and $\nu$ the $2$-isometries obtained as multiplication by $z$ on $D(\mu )$ and $D(\nu )$ are unitarily equivalent if and only if $\mu = \nu$. We also investigate similarity and quasisimilarity of these $2$-isometries, and we apply our results to the invariant subspaces of the Dirichlet shift.