Partially isometric Toeplitz operators on the polydisc. Issue 4 (20th April 2022) - K. D., Deepak; Pradhan, Deepak Kumar; Sarkar, Jaydeb
Abstract: A Toeplitz operator T φ $T_φ$, φ ∈ L ∞ ( T n ) $φ \in L^∞ (\mathbb {T}^n)$, is a partial isometry if and only if there exist inner functions φ 1, φ 2 ∈ H ∞ ( D n ) $φ _1, φ _2 \in H^∞ (\mathbb {D}^n)$ such that φ 1 $φ _1$ and φ 2 $φ _2$ depends on different variables and φ = φ ¯ 1 φ 2 $φ = \bar{φ }_1 φ _2$ . 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 ) $H^∞ (\mathbb {D}^n)$ . 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 ) $L^∞ (\mathbb {T}^n)$, 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 $T_φ$ is a shift whenever φ $φ$ is inner in H ∞ ( D n ) $H^∞ (\mathbb {D}^n)$ .