Partially isometric Toeplitz operators on the polydisc

Abstract

A Toeplitz operator $T_{\phi }$, $\phi \in L^{\infty }({\mathbb{T}}^n)$, is a partial isometry if and only if there exist inner functions ${\phi }_1,{\phi }_2\in H^{\infty }({\mathbb{D}}^n)$ such that ${\phi }_1$ and ${\phi }_2$ depends on different variables and $\phi ={\bar{\phi }}_1{\phi }_2$. In particular, for $n=1$, along with new proof, this recovers a classical theorem of Brown and Douglas. \noindent We also prove that a partially isometric Toeplitz operator is hyponormal if and only if the corresponding symbol is an inner function in $H^{\infty }({\mathbb{D}}^n)$. Moreover, partially isometric Toeplitz operators are always power partial isometry (following Halmos and Wallen), and hence, up to unitary equivalence, a partially isometric Toeplitz operator with symbol in $L^{\infty }({\mathbb{T}}^n)$, $n>1$, is either a shift, or a co-shift, or a direct sum of truncated shifts. Along the way, we prove that $T_{\phi }$ is a shift whenever $\phi$ is inner in $H^{\infty }({\mathbb{D}}^n)$.

12 pages. To appear in Bulletin of the London Mathematical Society