Czechoslovak Mathematical Journal, online first, 16 pp.

# Separately radial and radial Toeplitz operators on the projective space and representation theory

## Raul Quiroga-Barranco, Armando Sanchez-Nungaray

#### Received June 8, 2016.   First published March 1, 2017.

Raul Quiroga-Barranco, Centro de Investigación en Matemáticas, De Jalisco S-N, Valenciana, 36240 Guanajuato, Mexico, e-mail: quiroga@cimat.mx; Armando Sanchez-Nungaray, Facultad de Matemáticas, Universidad Veracruzana, Gonzalo Aguirre Beltrán, Isleta, 91090 Xalapa Enríquez, Veracruz, Mexico, e-mail: armsanchez@uv.mx

Abstract: We consider separately radial (with corresponding group ${\mathbb{T}}^n$) and radial (with corresponding group ${\rm U}(n))$ symbols on the projective space ${\mathbb{P}^n({\mathbb{C}})}$, as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the $C^*$-algebras generated by each family of such Toeplitz operators are commutative (see R. Quiroga-Barranco and A. Sanchez-Nungaray (2011)). We present a new representation theoretic proof of such commutativity. Our method is easier and more enlightening as it shows that the commutativity of the $C^*$-algebras is a consequence of the existence of multiplicity-free representations. Furthermore, our method shows how to extend the current formulas for the spectra of the corresponding Toeplitz operators to any closed group lying between ${\mathbb{T}}^n$ and ${\rm U}(n)$.

Keywords: Toeplitz operator; projective space

Classification (MSC 2010): 47B35, 32A36, 22E46, 32M15

DOI: 10.21136/CMJ.2017.0293-16

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References:
[1] M. Dawson, G. Ólafsson, R. Quiroga-Barranco: Commuting Toeplitz operators on bounded symmetric domains and multiplicity-free restrictions of holomorphic discrete series. J. Funct. Anal. 268 (2015), 1711-1732. DOI 10.1016/j.jfa.2014.12.002 | MR 3315576 | Zbl 1320.47029
[2] M. Engliš: Density of algebras generated by Toeplitz operators on Bergman spaces. Ark. Mat. 30 (1992), 227-243. DOI 10.1007/BF02384872 | MR 1289753 | Zbl 0784.46036
[3] R. Goodman, N. R. Wallach: Symmetry, Representations, and Invariants. Graduate Texts in Mathematics 255, Springer, New York (2009). DOI 10.1007/978-0-387-79852-3 | MR 2522486 | Zbl 1173.22001
[4] S. Grudsky, A. Karapetyants, N. Vasilevski: Toeplitz operators on the unit ball in $\C^n$ with radial symbols. J. Oper. Theory 49 (2003), 325-346. MR 1991742 | Zbl 1027.32010
[5] S. Grudsky, R. Quiroga-Barranco, N. Vasilevski: Commutative $C^*$-algebras of Toeplitz operators and quantization on the unit disk. J. Funct. Anal. 234 (2006), 1-44. DOI 10.1016/j.jfa.2005.11.015 | MR 2214138 | Zbl 1100.47023
[6] M. A. Morales-Ramos, A. Sánchez-Nungaray, J. Ramírez-Ortega: Toeplitz operators with quasi-separately radial symbols on the complex projective space. Bol. Soc. Mat. Mex., III. Ser. 22 (2016), 213-227. DOI 10.1007/s40590-015-0073-7 | MR 3473758 | Zbl 06562396
[7] R. Quiroga-Barranco: Separately radial and radial Toeplitz operators on the unit ball and representation theory. Bol. Soc. Mat. Mex., III. Ser. 22 (2016), 605-623. DOI 10.1007/s40590-016-0111-0 | MR 3544156 | Zbl 06646397
[8] R. Quiroga-Barranco, A. Sanchez-Nungaray: Commutative $C^*$-algebras of Toeplitz operators on complex projective spaces. Integral Equations Oper. Theory 71 (2011), 225-243. DOI 10.1007/s00020-011-1897-9 | MR 2838143 | Zbl 1251.47065
[9] R. Quiroga-Barranco, N. Vasilevski: Commutative $C^*$-algebras of Toeplitz operators on the unit ball, I.: Bargmann-type transforms and spectral representations of Toeplitz operators. Integral Equations Oper. Theory 59 (2007), 379-419. DOI 10.1007/s00020-007-1537-6 | MR 2363015 | Zbl 1144.47024
[10] R. Quiroga-Barranco, N. Vasilevski: Commutative $C^*$-algebras of Toeplitz operators on the unit ball, II.: Geometry of the level sets of symbols. Integral Equations Oper. Theory 60 (2008), 89-132. DOI 10.1007/s00020-007-1540-y | MR 2380317 | Zbl 1144.47025