Concrete representation of the V-Hankel dilation operator

Construct an explicit, concrete representation for the V-Hankel operator Y on K that arises from a contraction T on a Hilbert space H with minimal isometric dilation V on K and a bounded operator X: H → H satisfying T X = X T, where Y must satisfy V Y = Y V and Y|_H = X with respect to the orthogonal decomposition K = H ⊕ (K ⊖ H).

Background

Section 5.2 extends the lifting perspective to the setting of dilation theory. Given a contraction T on H and its minimal isometric dilation V on a larger Hilbert space K, an operator Y on K is termed V-Hankel if it satisfies the intertwining relation VY = YV.

Proposition 5.2 establishes that for any bounded operator X on H with TX = XT, there exists a V-Hankel operator Y on K such that the compression of Y to H equals X (i.e., Y|_H = X), with respect to the decomposition K = H ⊕ (K ⊖ H). However, the authors note that they do not have a concrete formula or canonical construction for such a Y, leaving its explicit representation unresolved.