Page 290 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Richa Nandra, Lovely Professional University Unit 27: The Adjoint of an Operator

Unit 27: The Adjoint of an Operator Notes

CONTENTS
Objectives

Introduction
27.1 Adjoint of an Operator
27.2 Summary

27.3 Keywords
27.4 Review Questions
27.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Define the adjoint of an operator.
 Understand theorems on adjoint of an operator.

 Solve problems on adjoint of an operator.
Introduction

We have already proved that T gives rise to an unique operator T* and H* such that (T*f) (x) =
f(Tx) f H * and x H. The operator T* on H* is called the conjugate of the operator T on H.

In the definition of conjugate T* of T, we have never made use of the correspondence between H
and H*. Now we make use of this correspondence to define the operator T* on H called the
adjoint of T. Though we are using the same symbol for the conjugate and adjoint operator on H,
one should note that the conjugate operator is defined on H*, while the adjoint is defined on H.

27.1 Adjoint of an Operator

Let T be an operator on Hilbert space H. Then there exists a unique operator T* on H such that
(Tx,y)= (x,T*y) for all x, y H
The operator T* is called the adjoint of the operator T.

Theorem 1: Let T be an operator on Hilbert space H. Then there exists a unique operator T* on H
such that

(Tx,y)= (x,T*y) for all x, y H ...(1)
The operator T* is called the adjoint of the operator T.
Proof: First we prove that if T is an operator on H, there exists a mapping T* on H onto itself
satisfying
(Tx,y)= (x,T*y) for all x,y H. ...(2)

LOVELY PROFESSIONAL UNIVERSITY 283