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Exercises in
Functional Analysis II, winter 2016/17

This site provides material for the exercise classes complementing the lecture course Functional Analysis 2 held by Prof. Thomas Østergaard Sørensen, PhD.

News

24.02.2017 The re-exam has been corrected and graded. Grades will be posted on Monday, February 27, at 10am, outside my office (B333). You can see your exam copy on February 27 from 10-11 am in B333.
14.02.2017 The exam has been corrected and graded. Grades will be posted on Wednesday, February 15, at 10am, outside my office (B333). You can see your exam copy on February 15 from 10-11 am in B333.

The re-exam is scheduled to take place on Thursday, February 23, 10am-1pm. Please check back on the main page for further information.

13.01.2017 Information about the exam is online on this webpage. Please read carefully and sign up in due time following the instructions there.
22.12.2016 In addition to Assignment 10, a "Holiday Extra sheet" has been released today. It does not consist of real training exercises, but rather a guide through the proofs of an extension to the functional calculus theorems proved in the lecture, namely to several commuting s.a. operators and to normal operators.

Besides introducing these theorems, which will be needed in the continuation of the lecture, the extra sheet should serve as an invitation to actively review and work with the theory exposed in the lecture so far. There will be no systematic discussion of the solutions, but you are invited to ask your questions about this sheet in the exercise class or during office hours.

17.10.2016 Registration for the exercises via the Lecture Assistant is open from now. Please sign up if you plan to attend! (You do not commit to anything by registrating, it just helps to get an overview of the public.)
14.10.2016 An exercise sheet consisting of warm-up problems is online. It is mostly designed to review some concepts already introduced in Functional Analysis I. The solutions will be discussed in the first week's exercise session (!) on October 20.

Exercise class

The exercise class will take place each Thursday from 12 to 14 in room B 132. There we will discuss the homework assignment released in the previous week.

Your solutions to the exercise sheets need not (resp. cannot) be handed in for grading. No model solutions will be published here.

Exercise sheets

Release date
Homework
Last updated
Main topics
Discussion
Oct 14
Warm-up sheet
Quotient Banach spaces, orthogonal projections, a compact integral operator
Oct 20
Oct 21
Assignment 1
Compact operators: properties and (counter-)examples; a multiplication operator
Oct 27
Oct 27
Assignment 2
Orthogonal projections, multiplication operator on a sequence space, resolvent map
Nov 3
Nov 4
Assignment 3
Shift operator, projections, complemented subspaces, codimension
Nov 10
Nov 10
Assignment 4
Spectrum of self-adjoint operators and general multiplication operators; Weyl sequences
Nov 17
Nov 18
Assignment 5
Spectral mapping theorem for polynomials, operator square root via power series, unitary operators and related ones
Nov 24
Nov 24
Assignment 6
Hilbert-Schmidt operators, a non-compact integral operator, spectrum under compact perturbations
Dec 1
Dec 1
Assignment 7
Weyl sequences II, Min-max principle, an explicit diagonalization
Dec 8
Dec 8
Assignment 8
The $\leq$-relation for s.a. operators, commutation under measurable functional calculus, normal operators
Dec 15
Dec 15
Assignment 9
Operator monotonicity of the inverse, Stone's formula, discrete Laplacian
Dec 22
Dec 22
Assignment 10
Operator monotone functions, functional calculus on a multiplication operator, spectral projections
Jan 12
Dec 22
Holiday Extra
Functional calculus for commuting self-adjoint operators and for normal operators
none
Jan 12
Assignment 11
Operator convex functions, a unitary group of operators, cyclic vectors
Jan 19
Jan 22
Assignment 12
Unbounded multiplication operator, adjoint, cyclic vectors, von Neumann's theorem
Jan 26
Jan 26
Assignment 13
Absolutely continuous functions, momentum operator on various domains
Feb 2
Feb 2
Assignment 14
Dirichlet and Neumann Laplacian, unitary groups of operators, Stone's theorem
Feb 9