Page 301 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis Richa Nandra, Lovely Professional University

Notes Unit 28: Self Adjoint Operators

CONTENTS
Objectives

Introduction
28.1 Self Adjoint Operator
28.1.1 Definition: Self Adjoint

28.1.2 Definition: Positive operator
28.2 Summary
28.3 Keywords
28.4 Review Questions

28.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Define self adjoint operator.
 Define positive operator.
 Solve problems on self adjoint operator.

Introduction

The properties of complex number with conjugate mapping z z motivate for the introduction
of the self-adjoint operators. The mapping z z of complex plane into itself behaves like the
adjoint operation in H as defined earlier. The operation z z has all the properties of the
adjoint operation. We know that the complex number is real iff z z . Analogue to this
characterization in H leads to the motion of self-adjoint operators in the Hilbert space.

28.1 Self Adjoint Operator

28.1.1 Definition: Self Adjoint

An operator T on a Hilbert space H is said to be self adjoint if T*=T.
We observe from the definition the following properties:
(i) O and I are self adjoint  O* O and I* I

(ii) An operator T on H is self adjoint if
Tx,y x,Ty x,y H and conversely.

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