1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference schemes as operator equations. General formulations 135

5) Assuming A to be a non-self-adjoint operator subject to the inequality

0 0 0

A > I A with I > 0 and A* = A > 0, we derive for a solution to equation

(37) the esti1nate

( 44)

The identity (A y, y) = ( <p, y) yields the chain of the relations

0

1(Ay, y) < (Ay, y) = (1.f!, y) < 111.f!ll,4-i · llYll,4,

thereby providing the validity of ( 44).

6) For a solution to equation (37) we thus have

( 45)

0 1

llAYll < I 111.f!ll

under the following conditions:

0 0 0 0 0

A*=A>1A, A= A*> 0, AA= AA, I> 0.

0
It suffices to show that II A y 11 > 111 A y 11 and to involve in further reasoning
0 0
equation (37), giving llAYll = 111.f!ll· The conditions A> 1A and AA=
0
A A together imply the chain of the relations

llAYll^2 = (Ay, Ay) = (A(A^112 y), (A^112 y))

0 0

> 1(A(A1f2y), (A1f2y)) =1(AAy, y)

0 0 0 0 0

= I ( A ( A 1I2 y ) , ( A 1I2 y ) ) > 12 ( A ( A 1/2 y ) , ( A 1/2 y ) )

0
meaning llAYll > 1llAYll· Here we exploit the fact that the operators A
0 0
and A^1!^2 as well as the operators A and A^112 commute with each other.