Kreĭn space representation and Lorentz groups of analytic Hilbert modules

Indefinite Schwarz-Pick inequalities on the bidisk (application of Hilbert module theory)

“…This class is closely related to submodules of rank 3 (see Wu-S-Yang [15] and Yang [16]). Further, we define other two classes as follows:…”

“…Example 2.3. Further non-trivial examples of elements in S(D 2 ; 2, 1) related to the theory of Hilbert modules in H 2 can be obtained from Theorem 3.3 in Wu-S-Yang [15]. P(D 2 ; 2, 1) and Q(D 2 ; 2, 1) are closed under composition of elements in Q(D 2 ; 2, 1) in the following sense (cf.…”

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

“…After an analysis on the spectral picture of core operators, paper [102] gives the following classification of submodules. In an attempt to find invariants for congruent submodules with infinite rank core operator, the notions of Lorentz group and little Lorentz group for submodules are defined and studied by Wu, Seto and the author [90]. The converse of Proposition 9.7 is not true.…”

“…First of all, we have the following fact from [38]. A two variable version of this lemma is shown in [90]. Since the set of nonzero complex numbers C × is a "trivial" subgroup in (H ∞ ) −1 , we only look at subgroups in (H ∞ ) −1 /C × .…”

“…Consider the ideals J n = w n H ∞ (D), n ≥ 0. Although G(J n ) is a proper subgroup in G(J n−1 ) for each n, they are all isomorphic to each other ( [90]). Theorem 9.10.…”