On quotient modules of H²(𝔻ⁿ): essential normality and boundary representations

Abstract

Let D n be the open unit polydisc in C n , n ≥ 1, and let H 2 (D n ) be the Hardy space over D n . For n ≥ 3, we show that if θ ∈ H ∞ (D n ) is an inner function, then the n-tuple of commuting operators (C z1 , . . . , C zn ) on the Beurling type quotient module Q θ is not essentially normal, whereRudin's quotient modules of H 2 (D 2 ) 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.2000 Mathematics Subject Classification. 47A13, 47A20, 47L25, 47L40, 46L05, 46L06.

A Brief Survey of Operator Theory in H 2(𝔻2)

“…A detailed study about the two spaces is made in Bickel and Gorkin [11]. It is worth noting that in the polydisc case n ≥ 3, Das, Gorai and Sarkar [23] observed that Beurling type quotient modules H 2 (D n ) ⊖ θH 2 (D n ) is never essentially normal.…”