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

Quiroga-Barranco, Raul, and Sanchez-Nungaray, Armando. "Separately radial and radial Toeplitz operators on the projective space and representation theory." Czechoslovak Mathematical Journal 67.4 (2017): 1005-1020. <http://eudml.org/doc/294385>.

@article{Quiroga2017,
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 $\{\rm U\}(n)$.},
author = {Quiroga-Barranco, Raul, Sanchez-Nungaray, Armando},
journal = {Czechoslovak Mathematical Journal},
keywords = {Toeplitz operator; projective space},
language = {eng},
number = {4},
pages = {1005-1020},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Separately radial and radial Toeplitz operators on the projective space and representation theory},
url = {http://eudml.org/doc/294385},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Quiroga-Barranco, Raul
AU - Sanchez-Nungaray, Armando
TI - Separately radial and radial Toeplitz operators on the projective space and representation theory
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 4
SP - 1005
EP - 1020
AB - 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 ${\rm U}(n)$.
LA - eng
KW - Toeplitz operator; projective space
UR - http://eudml.org/doc/294385
ER -