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

Czechoslovak Mathematical Journal

Raul Quiroga-Barranco, Armando Sanchez-Nungaray¹ 

关键词: Toeplitz operator;    projective space;   

DOI  :  10.21136/CMJ.2017.0293-16

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

  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 ${\rm U}(n)$.

【 授权许可】

Unknown