2 1. UNBOUNDED LINEAR OPERATOR S
a restriction of S or S an extension of T, denoted T C 5 or 5 2 T, if £(T) C £(S)
and Tu = Su for all ix G V(T). Clearly T = 5 if and only if T C 5 and 5 C T.
The sum T + S is the linear operator from H to i?i defined by
V(T + S) = £(T) H £(S), (T + S)u = Tu + Su,
and for any scalar a the scalar multiple aT is the linear operator from H to Hi
defined by
V(aT) = V(T), {aT)u = a(Tu).
If U is a linear operator from H\ to #2, then the product UT is the linear operator
from H to H2 defined by
V(UT) = {ue V(T) I Tu G V{U)}, {UT)u = U(Tu).
Finally, if T is 1-1, then the inverse
T~1
is the linear operator from to H defined
by
V(T-1)=7l{T)
with T~xu v \iTv u. In working with unbounded operators, one always has
to pay special attention to the domains of the operators.
To each densely defined linear operator from H to H\ there corresponds an-
other linear operator from H\ to H called the adjoint. Adjoint operators play an
important role in the theory of unbounded linear operators. To introduce the ad-
joint operator, suppose T is a linear operator from H to Hi with V{T) = H. Let
V(T*) denote the set of all v G H\ such that there exists v* G H with
(1.1) (Tu,v) = (u,v*) for all u G £(T).
Clearly V{T*) is nonempty since we can use v 0, v* = 0. Also, if
(u, v*) - (Tu, v) = (ix,^**) for all u G P(T),
then (u,v* v**) = 0 for all u G T(T), and the denseness of V(T) implies that
v* = v**. Consequently, for each v G V(T*) the element v* appearing in (1.1) is
uniquely determined. Let T*: V(T*) ^ H be the function defined by
T*v = v\
T* is called the adjoint of T. Note that equation (1.1) can be rewritten as
(1.2) (Tu,v) = (u,T*v) for all u G £(T), v G P(T*),
and in case T G B(H,Hi), this concept of an adjoint coincides with the usual
concept of a Hilbert space adjoint. It is simple to check that T* is a linear operator
from Hi to H.
The next four theorems list some of the elementary properties of adjoint oper-
ators.
THEOREM
1.2. If T and S are linear operators from H to Hi with T C S and
V(XJ = H, thenS* C f .
THEOREM
1.3. If T is a linear operator from H to Hi with V(T) = H and
V(T*) = Hlf thenTCT**.
Previous Page Next Page