Abstract :
Let T1 :H1 → H1 be a completely non-unitary contraction having a 2 × 2 singular
characteristic function Θ1; that is, Θ1 = [θi, j ]i, j=1,2 with θi j ∈ H∞ and det(Θ1) = 0. As
it is well known, Θ1 is a singular matrix if and only if Θ1 can be written as Θ1 = w1m1 a1
b1 [c1 d1 ] where w1,m1,a1, b1, c1,d1 ∈ H∞ are such that (i) w1 is an outer
function with |w1| 1, (ii) m1 is an inner function, (iii) |a1|2 + |b1|2 = |c1|2 + |d1|2 = 1,
and (iv) a1 ∧ b1 = c1 ∧ d1 = 1 (here ∧ stands for the greatest common inner divisor). Now
consider a second completely non-unitary contraction T2 :H2 →H2 having also a 2 × 2
singular characteristic function Θ2 = w2m2 a2
b2 [c2 d2 ]. We give necessary and sufficient
conditions for T1 and T2 to be quasi-similar
