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Richa Nandra, Lovely Professional University Unit 29: Normal and Unitary Operators

Unit 29: Normal and Unitary Operators Notes

CONTENTS
Objectives

Introduction
29.1 Normal and Unitary Operators
29.1.1 Normal Operator

29.1.2 Unitary Operator
29.1.3 Isometric Operator
29.2 Summary
29.3 Keywords

29.4 Review Questions
29.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand the concept of Normal and Unitary operators.
 Define the terms Normal, Unitary and Isometric operator.
 Solve problems on normal and unitary operators.

Introduction

An operator T on H is said to be normal if it commutes with its adjoint, that is, if TT*=T*T. We
shall see that they are the most general operators on H for which a simple and revealing
structure theory is possible. Our purpose in this unit is to present a few of their more elementary
properties which are necessary for our later work. In this unit, we shall also study about Unitary
operator and Isometric operator.

29.1 Normal and Unitary Operators

29.1.1 Normal Operator

Definition: An operator T on a Hilbert space H is said to be normal if it commutes with its
adjoint i.e. if TT* = T*T

Conclusively every self-adjoint operator is normal. For if T is a self adjoint operator i.e. T*=T
then TT* =T*T and so T is normal.

Note A normal operator need not be self adjoint as explained below by an example.

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