essential normality
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2019 ◽  
Vol 150 (3) ◽  
pp. 1339-1359 ◽  
Author(s):  
B. Krishna Das ◽  
Sushil Gorai ◽  
Jaydeb Sarkar

AbstractLet 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 $${\rm {\cal Q}}_\theta = H^2({\rm {\open D}}^n)/\theta H^2({\rm {\open D}}^n)\quad {\rm and}\quad C_{z_j} = P_{{\rm {\cal Q}}_\theta }M_{z_j}\vert_{{\rm {\cal Q}}_\theta }\quad (j = 1, \ldots ,n).$$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.


2016 ◽  
Vol 82 (12) ◽  
pp. 65-100
Author(s):  
Liangying Jiang ◽  
Caiheng Ouyang ◽  
Ruhan Zhao

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