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
(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
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
[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 * * .
1.11 (Stone [40]). Let T be the linear operator in
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
Since the continuous functions are dense
and each continuous function can be approximated uniformly by step
functions, it follows that the step functions are dense in
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
as n oo. This establishes the assertion.
Note that T0* is the zero operator defined on all of
To determine the adjoint T*, let S be the linear operator in
defined by
©(S) = C1[0,1], Su = u'.
Previous Page Next Page