fa.functional analysis – lifting theorem for n case

I am aware of the following statement of the lifting theorem. For i∈{1,2} let Bi be a contraction on a Hilbert space Hi and let Ai, acting on the Hilbert space Ki, be the minimal unitary dilation of Bi. Let Pi be the orthogonal projection of Ki onto Hi. Then an operator X from H1 to H2 satisfies B2X=XB1 if and only if there exist an operator Y from K1 to K2 such that

A2Y=YA1,
∥X∥=∥Y∥,
P2YP1=XP1.
I am looking for a similar theorem but for i∈{1,2,…n}. So now X satisfies n operator equations and I want a lift Y of X which will further satisfy n equations. Any reference /suggestion are most welcome for the above question.