Classes of stable two-layer schernes 405The outcome of further substitutions x = B-^1 Ay or y = A-^1 Bx is the
same:
(23) J A [y l = JA [ x l = 2 T [ ( B x , x) - ~ T (A x' x) l.
This implies that the conditions II S llA < 1 and B > ~TA are equivalent,
providing support for the view that Theorem 1 holds.
Since A= A* > 0, there exists a square root A^112 = (A^112 )* > 0, by
means of which the expression for J A [y] can be recast as
JA[Y] = (A1f2y,A1f2y) _ (A1/2 Sy,A1f2Sy)= llA^112 Yll^2 -llA^112 Syll^2
=II A^112 Y 112 - II (E-TC) A^112 Yll^2 '
where C = A^112 B-^1 A^112. Substitution u = A^112 y gives
1 A [y] = 11 u 112 - 11 ( E - T C) 1l 112 ·This implies that the condition
(24) llE-TCll < 1
is equivalent to the relation B >~TA. No assmnption is inade here that B
is a self-adjoint operator.
It seems worthwhile giving the special case when
B = B* > 0, A= A* > 0,for which the equivalence of the conditions
(25) J s[y] > 0 and B > 0.5 TA
is certainly true. Since B = B > 0, there exists B^112 = (B^112 ) > 0 and
this property is valid for the operator
C=B1/2A-1B1/2, C=C*>O.
By applying successively the substitutions B-^1 /^2 Ay = x, C^112 x = 1l, C =
B^112 A-^1 B^112 and B-^112 u = v we arrive at the chain of the relations
Js[y] = 2~(Ay,y)-T^2 (Ay,B-^1 Ay)
:= 2T(B^112 x,A-^1 B^112 )-T^2 (x,x)
= 2 T ( C X, X) - T^2 ( X 1 X) = 2 T ( 1l 1 1l) - T^2 ( c-l 1l, 1l)
= 2T(u,u)-T^2 (B-^1!^2 AB-^1 l^2 u,u)
= 2T ((Bv,v)- ~ (Av,v)).