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'.