Quasi-similarity of contractions having 2 x 2 a singular characteristic function

Abstract

LetT1:H1→H1be a completely non-unitary contraction having a 2×2 singular characteristic function Θ1; thatis, Θ1=[θi,j]i, j=1,2withθij∈H∞and det(Θ1)=0. Asit is well known, Θ1is a singular matrix if and only ifΘ1can be written asΘ1=w1m1[a1b1][c1d1]wherew1,m1,a1,b1,c1,d1∈H∞are such that (i)w1is an outer function with|w1| 1, (ii)m1is 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 contractionT2:H2→H2having also a 2×2singular characteristic function Θ2=w2m2[a2b2][c2d2].We give necessary and sufficient conditions forT1andT2to be quasi-similar.