4 1. UNBOUNDED LINEAR OPERATOR S
which implies that S C T*.
To establish the reverse inclusion, we show that V(T*) C T(S). Take any
v G £(T*), and set i;* = T*v G
L2[0,1],
so
(1.3) (Tu, v) = (u, v*) for all u G V(T).
Substituting u = 1 into (1.3) yields
(1.4) / v*{t)dt = 0= [ v*(t)dt.
Jo Jo
Let
w(t) = - / v*(s)ds, 0 t 1.
Clearly it; G ^ [ 0 , 1 ] , w(0) = 0, w(l) = 0 by (1.4), and w' = -v*, and from the
above it follows that w G V(S) C £(T*) and
(1.5) Sw = T*w = ?;*.
Combining (1.3) and (1.5), we have
(Tu, v) = (u, v*) = {u, T*w) = (Tu, w) for all u G V{T).
Since 7£(T) = C[0,1] is dense in L2[0,1], we conclude that v = w G X(5). This
proves that T)(T*) C X(5) and establishes that the adjoint T* is given by
V{T*)
^{ueH1
[0,1] | u(0) = u(l) = 0}, T*w = -u' .
Starting with the adjoint T*, a similar argument shows that Af(T*) = {0},
ft(T*) = (1 )
±
, and the adjoint of T* is given by
£(T**) = iJ1[0,l], T**tz = i/.
For this example we have T C T** and T ^ T * * .
EXAMPLE
1.11 (Stone [40]). Let T be the linear operator in
L2[0,1]
that has
domain V(T) consisting of all functions u G C[0,1] that are of bounded variation
on [0,1] with v! G L2[0,1] and with T defined by Tu = v!. Let T0 denote the
restriction of T to A/"(T), so
p(T0) = AT(T), T0u = Tu - 0.
We assert that V(TQ) is dense in
L2[0,1].
Since the continuous functions are dense
in
L2[0,1]
and each continuous function can be approximated uniformly by step
functions, it follows that the step functions are dense in
Z/2[0,1].
Each step function is a linear combination of characteristic functions X[aj]
where 0 a b 1. Consider a function X[a^]. For n = 1,2,... we construct
functions (f)n as follows: (j)n{t) = 0 for 0 t a 1/n or 6+ 1/n t 1; (f)n(t) 1
for a t b; on [a 1/n, a] (j)n{t) increases continuously and monotonically from 0
to 1 and has a zero derivative a.e.; and on [6, b + 1/n] j)n{t) decreases continuously
and monotonically from 1 to 0 and has a zero derivative a.e.. The construction
of (j)n on the intervals [a 1/n, a] and [b, b + 1/n] uses Cantor functions. Clearly
(j)n G V(TQ) and fin X[a^] in
I/2[0,1]
as n oo. This establishes the assertion.
Note that T0* is the zero operator defined on all of
L2[0,1].
To determine the adjoint T*, let S be the linear operator in
L2[0,1]
defined by
©(S) = C1[0,1], Su = u'.
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