CHAPTER 1

Unbounded Linear Operators

1. Introduction

In studying ordinary or partial differential operators in the Hilbert spaces

L2[a,

b] or

L2(Q),

one of the major difficulties is that these operators are not every-

where defined, nor are they bounded. Consequently, their study requires working

in the setting of unbounded operators. In this chapter we will develop the basic

theory of unbounded linear operators in a Hilbert space. Since this material is

well-known and is included in the standard textbooks on functional analysis and

operator theory, we shall omit most of the proofs. Standard references are Dunford

and Schwartz [6], Goldberg [9], Kato [17], Schechter [35], and Taylor and Lay [42].

The proofs for the first two sections are also given in Locker [24], the author's

earlier monograph.

Throughout this chapter we let # , # i , i7

2

,... denote complex Hilbert spaces.

Unless stated otherwise, the inner products and norms are denoted by the symbols

( , ) and || ||. The space 8(11, Hi) is the Banach space consisting of all bounded,

everywhere defined, linear operators from H to Hi with the uniform operator norm

imi = sup \\Tu\\.

IH=i

The Banach algebra B(H,H) is denoted by B(H).

DEFINITION

1.1. A linear operator from H to Hi is a function T which has

domain V(T) contained in H and range TZ(T) contained in Hi and which satisfies

the properties:

(a) V(T) is a subspace of H,

(b) T(u + v) = Tu + Tv, T(au) = aTu for all u, v e V{T) and for all scalars a.

Suppose T is a linear operator from H to H\. Let us once and for all set forth

the basic terminology associated with T. In case V(T) = H, T is said to be a linear

operator on H to Hi. If H = Hi, then we say that T is a linear operator in H,

while if V(T) = H = Hi, we say that T is a linear operator on H. If the domain

V(T) is dense in H, then T is said to be densely defined. The null space of T is the

subspace of H given by

Af(T) = {ue V(T) I Tu = 0}.

The linear operator T is said to be bounded if the supremum

sup{||Tu|| I u e V(T) with IMI = 1}

is finite, which is equivalent to T being continuous at each point of its domain.

Otherwise, T is said to be unbounded.

Suppose T and S are two linear operators from H to H\. T is said to be equal

to 5, denoted T = S, if V(T) = V(S) and Tu = Su for all u G V(T). T is said to be

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http://dx.doi.org/10.1090/surv/073/01