Hyponormal block Toeplitz operators with finite rank self-commutators

M Abhınand; Raul Curto; Prasad Thankarajan

doi:10.33205/cma.1817244

Hyponormal block Toeplitz operators with finite rank self-commutators

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 $T_\varphi$ is hyponormal and $[T_\varphi^{\ast}, T_\varphi]$ is a finite rank operator if and only if there exists a finite Blaschke product $b$ in $\mathcal{E}(\varphi)$, where $$ \mathcal{E}(\varphi) := \{k \in H^\infty(\mathbb{T}): \left\|k\right\|_\infty \le 1 \textrm{ and } \varphi-k\cdot \bar{\varphi} \in H^\infty(\mathbb{T}) \}. $$ An analogous set $\mathcal{E}(\Phi)$ can be defined for a matrix-valued symbol $\Phi$. In the block Toeplitz operator case, we first establish that if a symbol $\Phi$ is in $L^\infty(\mathbb{T},M_n)$ and if $\mathcal{E}(\Phi)$ contains a constant unitary matrix $U$, then $T_\Phi$ 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 $\Phi \in H^{\infty}(\mathbb{T}, M_n)$ is such that $\Phi^{\ast}$ is of bounded type and $T_\Phi$ is hyponormal. Then $[T_\Phi^{\ast}, T_\Phi]$ is a finite rank operator if and only if there exists a finite Blaschke–Potapov product in $\mathcal{E}(\widetilde{\Phi})$, where $ \widetilde\Phi:=\breve{\Phi}^*$ ; and ; $\breve{\Phi}(e^{i\theta}):=\Phi(e^{-i\theta})$.

Keywords

References

M. Abhinand, R. E. Curto, I. S. Hwang, W. Y. Lee and T. Prasad: Subnormal block Toeplitz operators, J. d’Analyse Math., 155 (2025), 485–500.
M. Abhinand, R. E. Curto, I. S. Hwang, W. Y. Lee and T. Prasad: Subnormal and hyponormal Toeplitz operators with operator-valued symbols, Preprint 2025.
M. Abrahamse: Subnormal Toeplitz operators and functions of bounded type, Duke Math. J., 43 (1976), 597–604.
J. Bram: Subnormal operators, Duke Math. J., 22 (1955), 75–94.
A. Brown, P. R. Halmos: Algebraic Properties of Toeplitz operators, J. Reine Angew. Math., 213 (1964), 89–102.
M. Cafasso: Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies, Math. Phys. Anal. Geom., 11 (2008), 11–51.
J. B. Conway: The theory of subnormal operators, Math surveys and Monographs, vol. 36, Amer. Math. Soc., Providence (1991).
C. Cowen: Hyponormality of Toeplitz operators, Proc. Amer. Math. Soc., 103 (1988), 809–812.

C. Cowen, J. Long: Some subnormal Toeplitz operators, J. Reine Angew. Math., 351 (1984), 216–220.
R. E. Curto, I. S. Hwang and W. Y. Lee: Hyponormality and subnormality of block Toeplitz operators, Adv. Math., 230, (2012), 2094–2151.
R. E. Curto, I. S. Hwang and W. Y. Lee: Which subnormal Toeplitz operators are either normal or analytic?, J. Funct. Anal., 263 (98), (2012), 2333–2354.
R. E. Curto, I. S. Hwang and W. Y. Lee: Operator-valued rational functions, J. Funct. Anal., 283 (9) (2022), Article ID: 109640.
R. E. Curto, W. Y. Lee: Joint hyponormality of Toeplitz pairs, Mem. Amer. Math. Soc., (2001).
J. B. Conway, L. Yang: Some open problems in the theory of subnormal operators, Holomorphic Spaces MSRI Publications, 33 (1998), 201–209.
C. Gu: A generalization of Cowen’s characterization of hyponormal Toeplitz operators, J. Funct. Anal., 124 (1) (1994), 135–148.
C. Gu, J. Hendricks and D. Rutherford: Hyponormality of block Toeplitz operators, Pacific J. Math., 223 (2006), 95–111.
P. R. Halmos: Normal dilations and extension of operators, Summa Brasil. Math., 2 (1950), 125–134.
P. R. Halmos: Ten problems in Hilbert space, Bull. Amer. Math. Soc., 76 (1970), 887–933.
P. R. Halmos: Ten years in Hilbert space, Int. Eq. Op. Theory, 2 (1979), 529–564.
M. Hayashi, F. Sakaguchi: Subnormal operators regarded as generalized observables and compound-system-type normal extension related to su(1,1), J. Phys. A: Math. Gen., 33 (2000), 7793–7820.
E. K. Ifantis: Minimal uncertainty states for bounded observables, J. Math. Phys., 12 (12) (1971), 2512–2516.
R. A. Martínez-Avendaño and P. Rosenthal: An introduction to operators on Hardy Hilbert space, Springer, New York (2007).
B. B. Morrel: A decomposition for some operators, Indiana Univ. Math., 23 (1973), 495–511.
T. Nakazi, K. Takahashi: Hyponormal Toeplitz operators and extremal problems of Hardy spaces, Trans. Amer. Math. Soc., 338 (1993), 753–769.
E. de Prunele: Conditions for bound states in a periodic linear chain, and the spectra of a class of Toeplitz operators in terms of polylogarithm functions, J. Phys. A: Math. Gen., 36 (2003), 8797–8815.
F. H. Szafraniec: Subnormality in the quantum harmonic oscillator, Commun. Math. Phys., 210 (2000), 323–334.
D. Xia: The analytic model of a subnormal operator, Int. Eq. Op. Theory, 10 (1987), 258–289.
D. Xia: Analytic theory of subnormal operators, Int. Eq. Op. Theory, 10 (1987), 880–903.

Details

Primary Language

English

Subjects

Operator Algebras and Functional Analysis

Journal Section

Research Article

Authors

M Abhınand
0000-0003-0923-9395
India

Raul Curto ^*
0000-0002-1776-5080
United States

Prasad Thankarajan
0000-0002-9237-6018
India

Publication Date

March 6, 2026

Submission Date

November 4, 2025

Acceptance Date

February 28, 2026

Published in Issue

Year 2026 Volume: 9 Number: 1

DOI

https://doi.org/10.33205/cma.1817244

IZ

https://izlik.org/JA33CJ24CB

Cite

RIS / Bibtex

APA

Abhınand, M., Curto, R., & Thankarajan, P. (2026). Hyponormal block Toeplitz operators with finite rank self-commutators. Constructive Mathematical Analysis, 9(1), 19-30. https://doi.org/10.33205/cma.1817244

AMA

1.Abhınand M, Curto R, Thankarajan P. Hyponormal block Toeplitz operators with finite rank self-commutators. CMA. 2026;9(1):19-30. doi:10.33205/cma.1817244

Chicago

Abhınand, M, Raul Curto, and Prasad Thankarajan. 2026. “Hyponormal Block Toeplitz Operators With Finite Rank Self-Commutators”. Constructive Mathematical Analysis 9 (1): 19-30. https://doi.org/10.33205/cma.1817244.

EndNote

Abhınand M, Curto R, Thankarajan P (March 1, 2026) Hyponormal block Toeplitz operators with finite rank self-commutators. Constructive Mathematical Analysis 9 1 19–30.

IEEE

[1]M. Abhınand, R. Curto, and P. Thankarajan, “Hyponormal block Toeplitz operators with finite rank self-commutators”, CMA, vol. 9, no. 1, pp. 19–30, Mar. 2026, doi: 10.33205/cma.1817244.

ISNAD

Abhınand, M - Curto, Raul - Thankarajan, Prasad. “Hyponormal Block Toeplitz Operators With Finite Rank Self-Commutators”. Constructive Mathematical Analysis 9/1 (March 1, 2026): 19-30. https://doi.org/10.33205/cma.1817244.

JAMA

1.Abhınand M, Curto R, Thankarajan P. Hyponormal block Toeplitz operators with finite rank self-commutators. CMA. 2026;9:19–30.

MLA

Abhınand, M, et al. “Hyponormal Block Toeplitz Operators With Finite Rank Self-Commutators”. Constructive Mathematical Analysis, vol. 9, no. 1, Mar. 2026, pp. 19-30, doi:10.33205/cma.1817244.

Vancouver

1.M Abhınand, Raul Curto, Prasad Thankarajan. Hyponormal block Toeplitz operators with finite rank self-commutators. CMA. 2026 Mar. 1;9(1):19-30. doi:10.33205/cma.1817244