Page 336 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Richa Nandra, Lovely Professional University Unit 31: Finite Dimensional Spectral Theory

Unit 31: Finite Dimensional Spectral Theory Notes

CONTENTS
Objectives
Introduction

31.1 Finite Dimensional Spectral Theory
31.1.1 Linear Operators and Matrices on a Finite Dimensional Hilbert Space
31.1.2 Similar Matrices

31.1.3 Determinant of an Operator
31.1.4 Spectral Analysis
31.1.5 Spectrum of an Operator
31.1.6 Spectral Theorem
31.1.7 Spectral Resolution of an Operator

31.1.8 Compact Operators
31.1.9 Properties of Compact Operators
31.2 Summary

31.3 Keywords
31.4 Review Questions
31.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand finite dimensional spectral theory.
 Describe spectral analysis and spectral resolution of an operator.
 Define compact operators and understand properties of compact operators.

 Solve problems on spectral theory.
Introduction

The generalisation of the matrix eigenvalue theory leads to the spectral theory of operators on
a Hilbert space. Since the linear operators on finite dimensional spaces are determined uniquely
by matrices, we shall study to some extent in detail the relationship between linear operators in
a finite dimension Hilbert spaces and matrices as a preliminary step towards the study of
spectral theory of operators on finite dimensional Hilbert spaces.

LOVELY PROFESSIONAL UNIVERSITY 329