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Richa Nandra, Lovely Professional University Unit 21: The Conjugate of an Operator

Unit 21: The Conjugate of an Operator Notes

CONTENTS
Objectives
Introduction

21.1 The Conjugate of an Operator
21.1.1 The Linear Function
21.1.2 The Conjugate of T

21.2 Summary
21.3 Keywords
21.4 Review Questions
21.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand the definition of conjugate of an operator.
 Understand theorems on it.

 Solve problems relate to conjugate of an operator.
Introduction

We shall see in this unit that each operator T on a normed linear space N induces a corresponding
operator, denoted by T* and called the conjugate of T, on the conjugate space N*. Our first task is
to define T* and our second is to investigate the properties of the mapping T T*.

21.1 The Conjugate of an Operator

21.1.1 The Linear Function

Let N* be the linear space of all scalar-valued linear functions defined on N. Clearly the conjugate
space N* is a subspace of N*. Let T be a linear transformation T of N* into itself as follows:
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If f N , then T (f) is defined as
[T (f)]x = f (T (x))
Since f (T (x)) is well defined, T is a well-defined transformation on N .
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Theorem 1: Let T : N N be defined as
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[T (j)] x = f (T (x)), f N , then
(a) T (j) is a linear junction defined on N.
(b) T is a linear mapping of N into itself.
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