Partially isometric Toeplitz operators on the polydisc

Article Type

Research Article

Publication Title

Bulletin of the London Mathematical Society

Abstract

A Toeplitz operator (Formula presented.), (Formula presented.), is a partial isometry if and only if there exist inner functions (Formula presented.) such that (Formula presented.) and (Formula presented.) depends on different variables and (Formula presented.). In particular, for (Formula presented.), 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 (Formula presented.). 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 (Formula presented.), (Formula presented.), is either a shift, or a co-shift, or a direct sum of truncated shifts. Along the way, we prove that (Formula presented.) is a shift whenever (Formula presented.) is inner in (Formula presented.).

First Page

1350

Last Page

1362

DOI

10.1112/blms.12633

Publication Date

8-1-2022

Comments

Open Access, Green

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