Preprint
Hyponormal block Toeplitz operators with finite rank self-commutators
ArXiv.org
Cornell University
05/04/2026
DOI: 10.48550/arxiv.2605.02214
Abstract
In this paper we identify a large class of hyponormal block Toeplitz operators whose self-commutators are of finite rank. Recall that an operator Tφ is hyponormal and [T ∗ φ , Tφ] is a finite rank operator if and only if there exists a finite Blaschke product b in E(φ), where E(φ) := {k ∈ H∞(T) : ∥k∥∞ ≤ 1 and φ − k · ¯φ ∈ H∞(T)}. An analogous set E(Φ) can be defined for a matrix-valued symbol Φ. In the block Toeplitz operator case, we first establish that if a symbol Φ is in L∞(T, Mn) and if E(Φ) contains a constant unitary matrix U , then TΦ is normal. We then obtain a suitable converse, under a mild assumption on the symbol. Next, we provide a partial answer to a conjecture recently posed by R.E. Curto, I.S. Hwang and W.Y. Lee [10, Conjecture 6.1]. Concretely, assume that Φ ∈ H∞(T, Mn) is such that Φ∗ is of bounded type and TΦ is hyponormal. Then [T ∗ Φ, TΦ] is a finite rank operator if and only if there exists a finite Blaschke–Potapov product in E(eΦ), where eΦ := ˘Φ∗ and ˘Φ(eiθ ) := Φ(e−iθ ).
Details
- Title: Subtitle
- Hyponormal block Toeplitz operators with finite rank self-commutators
- Creators
- Mankunikuzhiyil AbhinandRaul E CurtoThankarajan Prasad
- Resource Type
- Preprint
- Publication Details
- ArXiv.org
- DOI
- 10.48550/arxiv.2605.02214
- ISSN
- 2331-8422
- Publisher
- Cornell University; Ithaca, New York
- Language
- English
- Date posted
- 05/04/2026
- Academic Unit
- Mathematics
- Record Identifier
- 9985161339002771
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