418 Stability Theory of Difference SchernesThe condition a-> a- 0 implies that(52)
B--A>sA-. - T -^1
2 -Indeed, for a- > a-c
- T -
B - 2 A= A-^1 +(a-- 0.5) TE
=sA-^1 +(1-s)A-^1 +(a--0.5)rE
We have taken into account here that A-^1 > E/llA!!. Substituting (52)
into (51) we obtainThe generalized Cauchy-Bunyakovski'f inequality and the s-inequality to-
gether yield
Substituting (54) into (53) we arrive at the relation
or
.
Summing the preceding over k = 0, 1, 2, ... leads to estimate (50).
Lemma 5 Let A be a positive operator for which the inequality
( 55) 11 Ax 112 < 6. (Ax, x) ,
is valid with 6. = const > 0. Then
(56)
A-^1 > -^1 E
- and A<6.E.