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