1549301742-The_Theory_of_Difference_Schemes__Samarskii

410 Stability Theory of Difference Schemes

thereby justifying coinciding of the signs of the opera.tors C - / E a.nd

E - ~fc-^1. We now insert A-^1!^2 BA-^1!^2 in place of the inverse c-^1 a.nd

accept v = A-^1!^2 y. The outcome of this is

(Cx,x)-1(x,x) = (y,y)-1(A-^1 l^2 BA-^1 l^2 y,y) = (Av,v) -1(Bv,v),

meaning that the operators C -1 E a.nd A-1 B possess the same signs. By

merely setting / = / 1 a.nd / = / 2 we draw the conclusion that inequalities

(33) a.re equivalent.

Lemma 4 If the operator C = C* > 0 and T > 0, then the conditions

(34) II s II = II E - Tc II < p J

(35) l-p E<C< l+p E

T T

are equivalent.

Indeed, since the opera.tor S = E - rC = S* is self~a.djoint,

llSll= sup l(Sx,x)I= sup l((E-rC)x,x)I,

11x11= 1 11x11= 1

so that

or
-pE < S < pE

a.nd

• pE < E - rC < pE,

we deduce that
l-p E<C< l+p E.
T T

In view of this, condition (34) implies condition (35). The converse impli-
cation is simple to follow.

Theorem 3 Let A and B be positive operators and A= A, B = B > 0.
Then the conditions

1-o l+p
(36) --'B<A< B
T T