ECTS - Operator Theory
Operator Theory (MATH658) Course Detail
| Course Name | Course Code | Season | Lecture Hours | Application Hours | Lab Hours | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| Operator Theory | MATH658 | Area Elective | 3 | 0 | 0 | 3 | 5 |
| Pre-requisite Course(s) |
|---|
| N/A |
| Course Language | English |
|---|---|
| Course Type | Elective Courses |
| Course Level | Ph.D. |
| Mode of Delivery | Face To Face |
| Learning and Teaching Strategies | Lecture, Discussion, Question and Answer. |
| Course Lecturer(s) |
|
| Course Objectives | This is an introductory course in Operator Theory and its applications. We will start with a short review of Banach and Hilbert Spaces and their linear bounded operators, then we turn to the Spectral Theory of linear bounded operators. Classes of Hilbert Space operators that appear frequently in applications, e.g. self adjoint, positive operators, are considered in detail. The course is aimed at Mathematics students who want to pursue a career in Analysis and its applications, but Math students from other domains and also graduate engineering students who want to understand mathematical foundations of many of the subjects considered in Engineering Mathematics are welcome as well. |
| Course Learning Outcomes |
The students who succeeded in this course;
|
| Course Content | This is an introductory course in Operator Theory and its applications. We will start with a short review of Banach and Hilbert Spaces and their linear bounded operators, then we turn to the Spectral Theory of linear bounded operators. Classes of Hilbert Space operators that appear frequently in applications, e.g. self adjoint, positive operators, are considered in detail. |
Weekly Subjects and Releated Preparation Studies
| Week | Subjects | Preparation |
|---|---|---|
| 1 | A Review of Normed and Banach Spaces | [1], 2.2, 2.6, 2.7 [2], 3.1 |
| 2 | Bounded operators on Normed Spaces | [1], 2.7 [2], 4.3 |
| 3 | A Review of Inner Product and Hilbert Spaces | [1], 3.1,3.3,3.4 [2], 2.2—2.5 |
| 4 | Hilbert Adjoint Operator | [1], 3.8,3.9 |
| 5 | Spectral Theory in Normed Spaces: Introduction | [1], 7.2 |
| 6 | Spectral Theory of Bounded Linear Operators | [1], 7.3 |
| 7 | Spectral Mapping Theorem | [1], 7.4 |
| 8 | Review and Midterm | |
| 9 | Spectral Theory of Bounded Self Adjoint Linear Operators | [1], 9.1 |
| 10 | Spectral Theory of Bounded Self Adjoint Linear Operators | [1], 9.2 |
| 11 | Positive Operators | [1], 9.3 |
| 12 | Square Roots of a Positive Operator | [1], 9.4 |
| 13 | Projection Operators | [1], 9.5 |
| 14 | Further Properties of Projections | [1], 9.6 |
| 15 | Further properties of Projections | [1], 9.6 |
| 16 | Review |
Sources
| Course Book | 1. E. Kreyszig, Introductory Functional Analysis with Applications, Wiley Clas. Lib. Ed, 1989. |
|---|---|
| 2. G. Chacón, H. Rafeiro, J. Vallejo, Functional Analysis, De Gruyter, 2017. |
Evaluation System
| Requirements | Number | Percentage of Grade |
|---|---|---|
| Attendance/Participation | - | - |
| Laboratory | - | - |
| Application | - | - |
| Field Work | - | - |
| Special Course Internship | - | - |
| Quizzes/Studio Critics | - | - |
| Homework Assignments | 3 | 30 |
| Presentation | 1 | 20 |
| Project | - | - |
| Report | - | - |
| Seminar | - | - |
| Midterms Exams/Midterms Jury | 1 | 20 |
| Final Exam/Final Jury | 1 | 30 |
| Toplam | 6 | 100 |
| Percentage of Semester Work | 70 |
|---|---|
| Percentage of Final Work | 30 |
| Total | 100 |
Course Category
| Core Courses | X |
|---|---|
| Major Area Courses | |
| Supportive Courses | |
| Media and Managment Skills Courses | |
| Transferable Skill Courses |
The Relation Between Course Learning Competencies and Program Qualifications
| # | Program Qualifications / Competencies | Level of Contribution | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | Alanında, bağımsız olarak, bir problem kurgulayabilir, çözüm yöntemi geliştirerek problemi çözebilir ve sonuçları değerlendirebilir | X | ||||
| 2 | Matematiğin temel alanlarında ve kendi uzmanlığı olarak seçtiği alanda gerekli alt yapıyı oluşturur. | X | ||||
| 3 | Matematik literatürünü ve özel olarak kendi araştırma konusu ile ilgili ulusal ve uluslararası güncel yayınları takip edebilir ve bunlardan kendi araştırma konusu ile ilgili olanları çalışmalarında kullanabilir | X | ||||
| 4 | Bilimsel etik değerleri ve kuralları dikkate alır ve mesleki ve toplumsal yaşamda kullanabilir | X | ||||
| 5 | Kendi çalışmalarının sonuçlarını veya belli bir konudaki güncel çalışmaları ve bulguları, çeşitli bilimsel toplantılarda topluluk önünde Türkçe ve İngilizce olarak sunabilir ve tartışmalara katılabilir. | X | ||||
| 6 | Gerek bireysel, gerek bir çalışma grubunun üyesi olarak çalışabilme becerisini geliştirir | X | ||||
| 7 | Yaratıcı ve eleştirel düşünme, problem çözme, özgün bir çalışma üretme becerisini geliştirir. Bilimsel gelişmeleri takip eder, özümsediği bilgilerin analiz, sentez ve değerlendirmesini yapabilir. | X | ||||
| 8 | Kazandığı bilgi, beceri ve yetkinlikleri yaşam boyu geliştirmeye açık olur. | X | ||||
| 9 | Alanında özümsediği bilgiyi ve problem çözme yeteneğini disiplinler arası çalışmalarda uygulayabilir; karşılaşılan problemleri matematiksel modellerle ifade ederek, matematiksel bakış açısı ile farklı çözüm yöntemleri önerir. | X | ||||
| 10 | Matematik temelli yazılımları, bilişim ve iletişim teknolojilerini bilimsel amaçlı kullanabilir. | X | ||||
ECTS/Workload Table
| Activities | Number | Duration (Hours) | Total Workload |
|---|---|---|---|
| Course Hours (Including Exam Week: 16 x Total Hours) | 16 | 3 | 48 |
| Laboratory | |||
| Application | |||
| Special Course Internship | |||
| Field Work | |||
| Study Hours Out of Class | 14 | 2 | 28 |
| Presentation/Seminar Prepration | 1 | 10 | 10 |
| Project | |||
| Report | |||
| Homework Assignments | 3 | 5 | 15 |
| Quizzes/Studio Critics | |||
| Prepration of Midterm Exams/Midterm Jury | 1 | 12 | 12 |
| Prepration of Final Exams/Final Jury | 1 | 12 | 12 |
| Total Workload | 125 | ||