Partially isometric Toeplitz operators on the polydisc. Issue 4 (20th April 2022)

• Record Type: Journal Article
• Authors: K. D., Deepak; Pradhan, Deepak Kumar; Sarkar, Jaydeb
• Is Part Of: Bulletin of the London Mathematical Society. Volume 54:Issue 4(2022)
• Page Start: 1350
• Page End: 1362
• Publication Date: 2022-04-20
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1112/blms.12633
• Languages: English
• ISSNs: 0024-6093

Abstract

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 ) $H^\infty (\mathbb {D}^n)$ .