Quasi-similarity of contractions having a 2 × 1 characteristic function

Let T1 ∈ B(H1) be a completely non-unitary contraction having a non-zero characteristic function Θ1 which is a 2 × 1 column vector of functions in H∞. As it is well-known, such a function Θ1 can be written as Θ1 = w1m1 [ a1 b1 ] where w1,m1, a1, b1 ∈ H∞ are such that w1 is an outer function with |w1| ≤ 1, m1 is an inner function, |a1| + |b1| = 1, and a1 ∧ b1 = 1 (here ∧ stands for the greatest common inner divisor). Now consider a second completely non-unitary contraction T2 ∈ B(H2) having also a 2 × 1 characteristic function Θ2 = w2m2 [ a2 b2 ] . We prove that T1 is quasi-similar to T2 if, and only if, the following conditions hold: 1. m1 = m2, 2. {z ∈ T : |w1(z)| < 1} = {z ∈ T : |w2(z)| < 1} a.e., and 3. the ideal generated by a1 and b1 in the Smirnov class N+ equals the corresponding ideal generated by a2 and b2. 1. Statement of the main theorem Can one characterize the quasi-similarity of contractions in terms of their characteristic functions? Quasi-similarity is an equivalence relation between Hilbert space bounded operators which, being weaker than similarity, still preserves many interesting features as the eigenvalues, the spectral multiplicity or the non-triviality of the lattice of invariant subspaces (see [1], [3], [6] and references therein). 2000 Mathematics Subject Classification: 47A05, 47A45.