By J. Topuszanski
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Extra info for An Intro to Symmetry and Supersym. in Quantum Field Theory
8 + (Y 9 , g) 181 ~ 0 whenever f € HO' 9 € H01, and a. and 8 are complex scalars. The latter condition is equivalent to the positiveness of the 2 x 2 matrix ( Af,f) (Bf , g) (9,Bfl) (Yg,g) . , that 53 Halmos 1(Bf , g) 12 ~ (Af ,f) • (Vg • g) . Various conditions expressible in tenns of A and B only follow from this inequality. One of them is ker Acker B , another form of which is ker /A c ker B. (If Af = 0 for some f in HO ' then (Bf,g) = 0 for all 9 in H01, and therefore Bf = O. ) These kernel inclusions imply that ran B* c A and ran B* c ran IA (recall that A is Hennitian).
Yes; the argument goes as follows: IIYBfl1 2 = IIB(l - A)f~2 = (B(l-A)f,B(l-A)f) = ( B*B (1 - A) f , (1 - A) f) = (P2(1 - A) f , (1 - A) f) = ~ P(l - A) f I 2 = H1 - A) Pf ~ 2 ~ III _AI12(P2f, f) = III _A11 2 ~BfI12. Consequence: Y can be extended to a bounded operator on the closure of ran B (a subspace of HOL). * No matter how Y is defined on ranL B (= ker B*) , the matrix (~~) will satisfy three of the four equations that characterize idempotence; but it still remains to define Y on ker B* suitably so as to satisfy the fourth equation, Y - y2 = BB*.
The diagonal entries of the matrix are positive by assumption; all that needs to be checked is that the detenninant is positive. :q2 '1IC*g\l2 (Af, f) • (CC*g, g) COROLLARY 1. ~ necessary and sufficient condition that IA be ~ right factor of B ~ that there exist ~ positive operator Y such that 1(Bf, g) 12 ~ (A f , f) • (Y 9 , g) . Corollary 1 is a very slight sharpening of some known range inclusion and factorization theorems; see [5, #59]. COROLLARY 2. l1. if A ~ 0 and ran B* c ran /if. The characterization of subpositive suboperators is the subtlest result so far, but the characterization of subprojections lies slightly deeper still.
An Intro to Symmetry and Supersym. in Quantum Field Theory by J. Topuszanski