1. INTRODUCTION 3
THEOREM
1.4. Let T be a linear operator from H to Hi with V(T) = H, let
S e B ( # , # i ) , let aeC, and let U e B(HUH2). Then
(a)
(T +
S)* =T*
+
S*,
(b) (aT)* = aT*,
(c) (UT)* =T*U*.
THEOREM
1.5. IfT is a 1-1 linear operator from H to Hi with V(T) = H and
K(T) = Hi, then T* is 1-1 and (T*)"1 = {T~l)\
In the next definition we introduce the symmetric and self-adjoint operators.
These are very important classes of operators that have been studied extensively.
Since our primary interest lies in the non-self-adjoint operators, we do little more
than mention them here.
DEFINITION
1.6. Let T be a linear operator in H with V(T) = H.
(a) T is said to be symmetric if T C T*, i.e., if (Tu,v) (u,Tv) for all
u, v e V(T).
(b) T is said to be self-adjoint if T = T*.
For these linear operators we have the following fundamental results.
THEOREM
1.7. Let T be a linear operator in H with V(T) = H. IfT is 1-1
and self-adjoint, then
V^"1)
= 1Z{T) H and T
_ 1
is self-adjoint.
THEOREM
1.8. If T is a linear operator from H to Hi with V{T) = H, then
T* is bounded.
COROLLARY
1.9 (Hellinger-Toeplitz). IfT is a symmetric linear operator in H
with V{T) = H, then T is bounded and self-adjoint.
Let us illustrate these ideas with some examples of differential operators in the
Hilbert space L2[0,1].
EXAMPLE
1.10. Consider the linear operator T in
L2[0,1]
defined by
P(T) = C1[0,1], Tu = u'.
The functions un(t) = tn, n = 1,2,..., belong to V(T), and
,2
||2
= J'
en
dt = ^ p \\Tun f
n2*2-2
dt
n
\\un\r = I t""'dt=-——,
||'i'txn|rf==/
n~t- -dt=-^-
and
\Tun\
2 n + l
2 n - l
1/2
cxo as n - oo.
Thus, T is an unbounded linear operator in
L2[0,1].
Clearly M{T) = (1), and it is simple to check that U{T) = C[0,1). Let
H1 [0,1] be the subspace of L2[0,1] consisting of all functions u which are absolutely
continuous on [0,1] with u' G L2[0,1], and let S be the linear operator in L2[0,1]
defined by
V(S)
^{ueH1
[0,1] I u(0) - u(l) = 0}, Su = -u'.
We assert that T* = S. If u G V(T) and v ^(S), then upon integrating by parts
we get
(Tu,v)= u\t)^dt = u{t)vjt)\ - u{t)v'(t)dt = (u,Sv),
Jo I ° Jo
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