44 Basic Concepts of the Theory of Difference Schemes(c) lower semibounded if(6) (Ax, x) > -c, II x 112 for any x EH,
where c. is a positive number;
(d) positive definite if(7) (Ax, x) > 8 11 x 112 for any x EH,
where 8 is a positive number.
For an arbitrary nonnegative operator A and x E H, the number
(A:r, x) is called the energy of the operator A. Further comparison of
operators A and B will be carried out by means of the energy. If ( (A -
B)x, x) > 0 for all x, we write A> B. In particular, inequalities (4)-(7)
can be replaced by the following operator inequalities:A> 0, meanmg (Ax, x) > 0,
A> 0, mean mg (Ax, x) > 0,
(8)
A> -c. E, meanmg (Ax, x) > -c. 11x11^2 ,A> 8E, meanmg (Ax, x) > 8 11 x 112 ,
where E stands for the identity operator leaving a vector x unchanged:
Ex= x.
It is straightforward to verify that the relation established on the set
of linear operators (Hr-+ H) possesses the following properties:(a) A> Band C >Dimply A+ C > B + D;
(b) A > 0 and ,\ > 0 imply >.A > O;
( c) A > B and B > C imply A > C;
(cl) If A> 0 and A-^1 exists, then A-^1 > 0.If A is a linear operator defined on H, then the operator A on H
subject to the condition (Ax, y) = (x, Ay), x, y E Ii, is called the adjoint
operator to A. If A is a linear bounded operator, then its own adjoint is
uniquely defined and falls within the category of linear bounded operators
with the norm 11 A 11 = 11 A 11 · A linear bounded operator A is called self-
adjoint if A = A, that is, (Ax, y) = (x, Ay) for all x, y EH. If A is an