Partially isometric Toeplitz operators on the polydisc

Abstract

A Toeplitz operator T_φ, φ ∈ L^∞(Tⁿ), is a partial isometry if and only if there exist inner functions φ₁, φ₂ ∈ H^∞(Dⁿ) such that φ₁ and φ₂ depends on different ariables and φ = ¯φ₁φ₂. In particular, for 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^∞(Dⁿ). 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^∞(Tⁿ), 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 is a shift whenever φ is inner in H^∞(Dⁿ).