Partially isometric Toeplitz operators on the polydisc

Abstract

A Toeplitz operator T φ $T_\varphi$ , φ ∈ L ∞ ( T n ) $\varphi \in L^\infty (\mathbb {T}^n)$ , is a partial isometry if and only if there exist inner functions φ 1 , φ 2 ∈ H ∞ ( D n ) $\varphi _1, \varphi _2 \in H^\infty (\mathbb {D}^n)$ such that φ 1 $\varphi _1$ and φ 2 $\varphi _2$ depends on different variables and φ = φ ¯ 1 φ 2 $\varphi = \bar{\varphi }_1 \varphi _2$ . In particular, for n = 1 $n=1$ , along with new proof, this recovers a classical theorem of Brown and Douglas. We also prove that a partially isometric Toeplitz operator is hyponormal if and only if the corresponding symbol is an inner function in H ∞ ( D n ) $H^\infty (\mathbb {D}^n)$ . 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 ∞ ( T n ) $L^\infty (\mathbb {T}^n)$ , n > 1 $n > 1$ , is either a shift, or a co-shift, or a direct sum of truncated shifts. Along the way, we prove that T φ $T_\varphi$ is a shift whenever φ $\varphi$ is inner in H ∞ ( D n ) $H^\infty (\mathbb {D}^n)$ .