Relevant elements of functional analysis 45arbitrary linear operator, then A* A and AA* are self-adjoint nonnegative
operators:(AAx,y) = (Ax,Ay) = (x,AAy), (A* Ax, x) = 11 Ax 112 > 0,
(AA x) x) = 11 A x 112 > 0 '
It is worth noting the obvious relations (A*)* =A and (A*)-^1 = (A-^1 )*.
Any nonnegative operator A in a complex Hilbert space I-I is self-
adjoint:- if (Ax,x) > 0 for all x EH, then A= A*.
For real Hilbert spaces this statement fails to be true. As far as only real
Hilbert spaces are considered, we will use the operator ineqnalities for non-
self-adjoint operators as well.
Theoren1 2 The pl'oduct AB of two commuting nonnegative self-adjoint
opel'atol's A and B is also a nonnegative self-adjoint opel'ato1-.
An operator B is called a square root of an operator A if B^2 = A.Theorem 3 The1·e exists a unique nonnegative self-adjoint squal'e poot B
of any nonnegative self-adjoint opel'ator A commuting with any opel'atol'
which commutes with A.
We denote by A^112 the square root of an operator A.
Let A be a positive self-adjoint linear operator. By introducing on
the space H the inner product ( x, y) A = (Ax, y) and the associated norm
11 x llA = J(x, x)A we obtain a Hilbert space HA, which is usually called
the energetic space HA. It is easy to show that the inner product
(x,y)A = (Ax,y)
satisfies all of the axioms of the inner product:
(1) (x,y)A = (y,x)A;
(2) (x + y, z)A = (x, z)A + (y, z)A;
(3) (-Xx, Y)A = -X(x, Y)A;
(4) (x, x)A > 0 for x f. 0 and (x, x) = 0 only for x = 0.
Axioms (2) and (3) are met by virtue of the linearity property. The
validity of ( 4) is stipulated by the fact that the operator A is positive. The
meaning of the self-acljointness of the operator A is that we should have