Beurling type quotient modules over the bidisk and boundary representations

Abstract

This paper mainly concerns Beurling type quotient modules of H 2 (D 2 ) over the bidisk. By establishing a theorem of function theory over the bidisk, it is shown that a Beurling type quotient module is essentially normal if and only if the corresponding inner function is a rational inner function having degree at most (1, 1). Furthermore, we apply this result to the study of boundary representations of Toeplitz algebras over quotient modules. It is proved that the identity representation of C * (S z , S w ) is a boundary representation of B(S z , S w ) in all nontrivial cases. This extends a result of Arveson to Toeplitz algebras on Beurling type quotient modules over the bidisk (cf. [W. Arveson, Subalgebras of C * -algebras, Acta Math. 123 (1969) 141-224; W. Arveson, Subalgebras of C * -algebras II, Acta Math. 128 (1972) 271-308]). The paper also establishes K-homology defined by Beurling type quotient modules over the bidisk.

Properties of Beurling-type submodules via Agler decompositions

“…Observe that this result complements the Guo and Wang result from [20] discussed earlier. In fact, Theorem 1.1 together with Guo-Wang's result implies that if S z 1 and S z 2 are simultaneously essentially normal, then the two commutators are actually at most rank one!…”

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

“…A generalization of this theorem to the Hardy space over the polydisc H 2 (D n ) also hold ( [18,89]). Theorem 7.2 (Guo and K. Wang [42]). Let θ ∈ H 2 (D 2 ) be an inner function.…”

Multiplication Operators on the Bergman Space

“…/ is called an analytic Hilbert module [CG,Guo1,Guo2,Guo3,Guo4,GW2,GW3,GW4] if the following hold: Now let be a domain in C d , and denote by A.…”