Relevant ele1nents of functional analysis 43
it is possible to introduce the operation of multiplication AB of operators
A and B: (AB)x = A(Bx). Clearly, AB is a linear bounded operator in
light of the obvious relation II AB II~ II A II · II B II· If (AB)x = (BA)x for
all x E X, then operators A and B are said to be commuting. In that
case we will write AB = BA.
While solving equations of the form Ax = y we shall need yet the
notion of the inverse A-^1. Let A be an operator from the space X into
the space Y. By definition, this means that D(A) = X and R(A) = Y.
If to each y E Y there corresponds only one element x E X, for which
Ax = y, then this correspondence specifies an operator A-^1 , known as the
inverse for A, with the domain Y and range X. By the definition of inverse
operator, we have for any x E .X. and any y E YIt is easy to show that if an operator A is linear, then so is the inverse A-^1
(if it exists).Lemma 1 In order that an additive operator A with D(A) = X and
R(A) = Y possess an inverse, it is necessary and sufficient that Ax = 0
only ifx = 0.Theore111 1 Let A be a linear operator from .X. into Y. In order that the
inverse operator A-^1 exist and be bounded, as an operator from Y into)(,
it is necessary and sufficient that there is a constant 15 > 0 such that for all
x EX(3) II Ax 112 2 15 II x (^111)
(II· 111 is the norm on the space X and 11·11 2 is the norm on the space Y).
lVIoreover, the estimate II A-^1 II~ l/15 is valid.
- Linear bounded operators in a real Hilbert space. Let H be a real
Hilbert space equipped with an inner product (x, y) and associated norm
11x11 = ~· We consider bounded linear operators defined on the space
H (D(A) = H). Before giving further motivations, it will be convenient to
introduce several definitions. We call an operator A
(a) nonnegative if(4) (Ax, x) 2 0 for all x E JI;
(b) positive if(5) (Ax, x) > 0 for all x EH except for x = O· J