Difference schemes as operator equations. General formulations 133where A is a linear bounded operator defined in a real Hilbert space H, <p
is given and y is sought in the space H.
We will assume that problem (37) is solvable for any right-hand sides
<p E H: there exists an operator A-^1 with the domain D( A-^1 ) = H. All
the constants below are supposed to be independent of h. In what follows
the space H is equipped with an inner product ( , ) and associated norm
II x II = J( x, x ). The writing A= A* > 0 means that A is a self-adjoint
positive operator. SetII y llA = V( Ay, y)' A=A*>O.
An a priori estimate depends on the nature of the subsidiary information
on the operator A.
Let H be a finite-dimensional space.
1) Consider first the simplest case when A is a non-self-adjoint positive
definite operator:(38) A > 8 E, 8 > 0 or (Ay, Y) > 8llYll^2 for any y EH,
where Eis the identity operator. Then the inverse A-^1 is bounded in norm
by the constant 1 / 8:(39)Indeed,0 < (Ay, y) - 8IIy112 = (A-^1 x, x) - 8 II A-^1 x 112
< llA-^1 xll ·llxll-8llA-^1 xll^2
=8llA-^1 xll(~llxll-llA-^1 xll), x=Ay,
implying that 11A-^1 x11 < 5-l 11 x II or 11A-^111 < 5-^1. Since y = A-^1 <p and
11y11 < 11 A-^1 11 · 11 'P 11, the solution to equation (37) admits the estimate
( 40)
1
llYll < bll'Pll for A>8E, 8>0.- The precise estimate appears to be useful:
for A=A*>O.