32 P. DEIFT, L. C. LI, AND C. TOMEI

(2.101), at elements A in the open subset of the finite codimensional submanifold g* fl

{A(z) : A(l) = A

r e g

(l) = diagonal} of g*. . In particular this is true of A = Areg(z) is

positive definite on S1 with A(l) diagonal. As we will see, this is enough for our purposes.

In order to solve the interpolating flow by a Symes-type factorization, we must deter-

mine the map 0 (see Appendix) in our case g_ with i2-matrix given by(1.37), (1.33) and

(1.34). Simple computations show that the ideals K± are given by g fl {X(X = oo) = 0}

respectively, so that g /K± is isomorphic to the diagonal matrices. Also 0(6) = —6 for

any diagonal matrix 5, and condition (A.28) becomes

g+(t, A = oo) = g.(t, A = oo) . (2.102)

We must also consider matrix factorization problems of the following form: suppose that

X(X) is a self-adjoint element in ^,

X(X)=X(XY , XeiR . (2.103)

Then for any real t,

e

('/2)X(oo)

e

tX{\)

e

-(t/2)X(oo)

i s a s m o o t

h , positive definite matrix-

valued function on S, with limit

e

-(t/2)X(oo) etX(X) e-{t/2)X(oo) _+ j (2.104)

as A — * oo. By standard matrix factorization theory (see, for example, [G-K]), there exists

a (unique) factorization

e-(t/2)X(oc) etX(X) e-U/2)X(oo) =§+(tiX)g_(t,\) (2.105)

with g±(t, •) analytic in {Re A 0}, {Re A 0} respectively, continuous and invertible in

{Re A 0}, {Re A 0} respectively, and satisfying

g±(t,oo) = I . (2.106)

Thus

g+(t,X) = e^x^g+(t,X)

(2.107)

^ ( ^ A ) ^ ^ ( ^ A ) e ^ 2 ^ - )