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Extra info for An Intro to Symmetry and Supersym. in Quantum Field Theory

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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.

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An Intro to Symmetry and Supersym. in Quantum Field Theory by J. Topuszanski

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