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IRnxn x= A(t)x. 36) where the time-reversal operation is considered with respect to t. Proof. ). By applying now 0 to both sides of the above equality one gets further from which By replacing now d T T T dt

We continue this procedure until all purely imaginary eigenvalues of Hare exhausted. b) Let>.. I2n )m-lal f:. O. (H) is J-neutral and R>. (H)+R_>.. (H) is nondegenerate, there exists a vector b1 E R_>.. (H) with With we obtain as in a) that am, ... , al and bm, ... 2. hn)1-lal, j E N, j 2: 2, j=2 where a2, ... , am are, as in a), uniquely defined by the conditions [aI, bjJ = 0 for 1:s:: j :s:: m - 1; this yields raj, bkJ = 0 for j + k :s:: m. Moreover, raj, bkJ for j + k = m + 1 we obtain = 0 for j + k 2: m + 2 and (_l)k-l[a m , hJ (_l)k-l[a m ,hJ = (_l)k-l.

Clearly, L 2 ,n(lR) = L 2,n(-00,T] EB L 2,n[T,(0) for every T E IR. We introduce the family of orthogonal projections P;' from L 2,n(lR) onto L 2,n[T, (0) by t< t 2: T, T, together with the family of complementary orthogonal projections Q~ from L 2 ,n onto L 2 ,n( -00, T], given by Q~ := I - P;'. Here I is the identity on L 2 ,n. 9. 1. An operator Q : L 2,m(lR) QP;' 2. An operator F: L 2,m(lR) ----+ ----+ L 2,n(lR) is said to be causal if = P;'QP;' , "IT E IR. L 2,n(lR) is said to be anti-causal if FQr; = Q~FQr;, "IT IR.