Spring 2007

110.640 - Spectral Theory
MTW 3:00  // NEB 12
Michael Goldberg

Office Hours:  Held in Krieger 313.
Fri. 12:00 - 2:00pm, or by appointment.
Office Phone: (410) 516-7406
Email: mikeg@math.jhu.edu

Textbook: A Short Course on Spectral Theory by William Arveson,
Springer Verlag, 2002.   ISBN 978-0387953007

We will cover the first three chapters of this book, with some additional lectures to fill in the Functional Analysis background that the textbook requires. Major topics to be addressed include:

  • The spectrum of an operator: definition and properties.
  • Maximal Ideals and the Gelfand map.
  • Operators on a Hilbert space.
  • The Spectral Theorem and normal functional calculus.
  • Compact and Fredholm operators, Fredholm Index Theorem.

Grading:   Based on completion of weekly homework assignments.

Homework: Assignments from the textbook will be posted weekly.

Due Jan. 31:   Chapter 1,   Problems 1.4, 2.2-2.4, 3.2

Due Feb. 7:   Chapter 1,   Problems 4.2, 4.3, 5.1, 5.2, 6.2, 6.3

Due Feb. 14:   Chapter 1,   Problems 7.2-7.4, 8.3, 8.4, 9.1

Due Feb. 21:   Chapter 1,   Problems 9.3, 10.1, 10.2, 11.1-3

Due Feb. 28:   Chapter 2,   Problems 1.3-1.6

Due Mar. 7:   Chapter 2,   Problems 2.8, 3.1-3.3

Due Mar. 21:   Chapter 2,   Problems 4.3-4.8

Due Mar. 28:   Chapter 2,   Problems 6.2, 7.2, 8.1-2   (Hint: see Chapter 1, exercise 4.2)

Due Apr. 11:   Chapter 2,   Problems 9.2, 9.3

Due Apr. 18:   Chapter 2,   Problems 10.1, 10.2.   Chapter 3, Problems 1.1, 1.2, 2.1

Due Apr. 25:   Chapter 3,   Problems 2.4, 3.2-3.4
Hint: Extend the Fredholm Alternative to sums of (Invertible) + (Compact)

Homework assignments are due in lecture on Wednesday, and will be returned the following Monday.

You are permitted, perhaps encouraged, to discuss homework problems with other students. This collaboration should not extend to the process of writing up solutions. The work that you turn in should be written by you, in your own words, without supervision or other well-meaning influence from anyone else.

Ethics Statement: Cheating is wrong. Cheating hurts our community by undermining academic integrity, creating mistrust, and fostering unfair competition. The university will punish cheaters with failure on an assignment, failure in a course, permanent transcript notation, suspension, and/or expulsion. Offenses may be reported to medical, law, or other professional or graduate schools when a cheater applies.

Violations can include cheating on exams, plagiarism, reuse of assignments without permission, improper use of the Internet and electronic devices, unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition. Ignorance of these rules is not an excuse.

In this course there should be very little incentive for students to engage in unethical behavior.

Medical Contingencies:

The Student Health Center recently adopted new guidelines for the issuance of written Medical Excuses. Please read this memorandum for more information. A one-sentence summary is that the Health Center will now only document serious and/or prolonged illnesses for which they have actively provided treatment.

Students with disabilities requiring accommodation should notify me as soon as possible so that we can make the appropriate arrangements.