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

Full text available as PDF.


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