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 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 ^ - )
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