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

Quiroga-Barranco, Raul ; Sanchez-Nungaray, Armando

Czechoslovak Mathematical Journal, Tome 67 (2017) no. 4, pp. 1005-1020.

Résumé

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 ${\rm U}(n)$.

Détail

DOI : 10.21136/CMJ.2017.0293-16

Classification : 22E46, 32A36, 32M15, 47B35
Keywords: Toeplitz operator; projective space