1 results
On quotient modules of H2(𝔻n): essential normality and boundary representations
- Part of
-
- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 3 / June 2020
- Published online by Cambridge University Press:
- 31 January 2019, pp. 1339-1359
- Print publication:
- June 2020
-
- Article
- Export citation
-
Let 𝔻n be the open unit polydisc in ℂn,
$n \ges 1$, and let H2(𝔻n) be the Hardy space over 𝔻n. For 
$n\ges 3$, we show that if θ ∈ H∞(𝔻n) is an inner function, then the n-tuple of commuting operators 
$(C_{z_1}, \ldots , C_{z_n})$ on the Beurling type quotient module 
${\cal Q}_{\theta }$ is not essentially normal, where
Rudin's quotient modules of H2(𝔻2) are also shown to be not essentially normal. We prove several results concerning boundary representations of C*-algebras corresponding to different classes of quotient modules including doubly commuting quotient modules and homogeneous quotient modules.
$${\rm {\cal Q}}_\theta = H^2({\rm {\open D}}^n)/\theta H^2({\rm {\open D}}^n)\quad {\rm and}\quad C_{z_j} = P_{{\rm {\cal Q}}_\theta }M_{z_j}\vert_{{\rm {\cal Q}}_\theta }\quad (j = 1, \ldots ,n).$$
