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This being true on a dense subset (and since AX is a contraction), the same statement must be true for all x ∈ X . Thus, Σ is strongly stable, as claimed. That Σ is strongly co-stable whenever Σ• is strongly co-stable is proved in a similar way. 6. Let θ be a B(U ; Y )-valued Schur function on the open unit disk D. Then the following conditions are equivalent. 1) θ has a bi-inner dilation. 2) θ has both an inner dilation and a co-inner dilation. 3) The balanced passive realization of θ is both strongly stable and strongly costable.
The latter inclusion follows trivially from the fact that R (B ) = R (B• ) ⊂ X• ⊂ X . 2 with Y◦ = X◦ and Y• = X• . ) 36 Arov and Staﬀans We next show that Σ following substitutions: IEOT is passive. 2 with the X◦ → X◦ ⊕ U, X• → X• ⊕ U, Y◦ → X◦ ⊕ Y, Y• → X• ⊕ Y, A◦ → A A◦ C◦ B◦ . D◦ B is contractive, hence Σ is passive. 2, C D To see that Σ is controllable we argue as follows. By construction, the reachability maps B• and B of Σ• respectively Σ have the same range. , R (B• ) is dense in X• . Since X• is dense in X , also R (B ) = R (B• ) is dense in X .
As an example, take exp(iλx), with λ > 0, and we will obtain a model function for the real line R. In the space L2 (Γ0 ) let us also consider the pair of complementary projections: Ph = hQΓ0 h−1 I, Qh = hPΓ0 h−1 I , and the subspace M(h) = Ph (L2+ (Γ0 )). Vol. 14. , n. Then h = n j=1 hj ∈ U and n−1 M(h) = M(h1 ) ⊕ h1 M(h2 ) ⊕ . . ⊕ ( hj )M(hn ) . j=1 Let ak ∈ C, k = 1, 2, 3, 4, and assume that ∆ = a1 a4 − a2 a3 = 0. Consider the following two fractional linear transformations, which are inverses of one another: a1 t + a2 a4 x − a2 v(t) = .