Relevant elernents of functional analysis^47which confirms the equivalence of (10) and (10').
In the sequel sufficient conditions for the existence of a bounded in-
verse operator A-^1 defined in the entire space H, D(A-^1 ) = H, will be of
great importance for us.
We note in passing that Lemma 1 and Theorem 1 guarantee the ex-
istence of an inverse operator defined only on R(A), the range of A, which
is not obliged to coincide with H. If the range of an operator A happens
to be the entire space H, R(A) = H, then the conditions of Lemma 1 or
Theoren1 1 ensure the existence of an operator A-^1 with D(A-^1 ) = H. In
particular, a positive operator A with the range R(A) = H possesses an
inverse A -I with D( A -I) = H, since the con di ti on (Ax, x) > 0 for all x f:- 0
implies that Ax f:- 0 for x f:- 0 and Lemma 1 applies equally well to such a
setting.Theorem 4 Let A be a linear bounded operator in Hilbert space H,
D( A) = H. In order that the operator A possess an inverse operator A-^1
with the domain D(A-^1 ) = H, it is necessary and sufficient the existence
of a constant 8 > 0 such that for all x E H the following inequalities hold:11A*x11 > 8 11 x II·
Moreover, the estimate II A-^1 II< 1/8 is true.
Corollary Let A be a positive definite linear bounded operator with the
domain D( A) = H. Then there exists a bounded in verse operator A-^1 with
the domain 'D(A-^1 ) = H.
Indeed, with the relation A > 8E, 8 > 0 in view, we arrive at the
chains of the relations11 Ax 11 11 x 11 > (Ax, x) > 8 11 x 112 ,
11 A x 11 11 x 11 > I (A x, x) I = I ( x, Ax) I = (Ax, x) > 811 x 112 ,
thereby providing the relations II Ax II> 811 x II and II A*x II> 811 x II as well
as the validity of the conditions of Theorem 4. The norm of the inverse
admits the estimate A-^1 < 1/8.
Remark For the existence of an inverse A-^1 in a finite-dimensional Hilbert
space it suffices to require the positiveness of the operator A, since the
condition A > 0 implies the existence of a constant 8 > 0 such that
(Ax, x) > 811x11^2