FORMAL REALIZ ABILITY THEORY — II 547 



ill K is non-singular. Dually 



i + z{v)y{p) 



in V is non-singular. 



Proof: Suppose that A; e K is such that 



(1 + Y{v)Z{v))k = (1) 



for all p. TluMi 



= Z(/))(l + Y{v)Z{v))k = Z{p)k -f Z{p)Y(p)Z(p)k 



for all p. Then, however, 



iZ(p)k, k) + (Z(p)Y{p)Z{p)k, k) = 0. (2) 



We may write the second term as 



{Z*(p)k, Y{p)Z(p)k) = {Z{p)k, Y*(p)Z*{p)k) (3) 



by (I, 14.0) applied t^^ice. Now Z(p) is PR, in particular real and sym- 

 metric, so 



Z*ip) = Z*(p) = Z'ip) = Zip). 



Using a similar calculation with Y(p), the quantity (3) becomes 



{Z{p)k, Y(p)Z(p)k). (4) 



For each p e T+ , we have p e r+ and the first term of (2) has a non- 

 negative real part. But for p e r+ , (4) is the conjugate of 



(v, Y(p)v) (5) 



where v = Z(p)k. Xow (5) is a PR function of p, hence has a non-nega- 

 Uve real part for p e T+ , for any v. In particular therefore this is true 

 for the V which, at p, makes (5) the conjugate of (4). Therefore (4) has 

 a non-negative real part throughout r+ . It follows from (2) then that 



Re{Z{p)k, k) = 



for all p er+. By 2.02, then, 



Z(p)k = 0. 

 By (1), then 



Ik = k = 0. 

 Hence (1) implies k = 0. Therefore the operator in (1) has an inverse 



