exercises and tutorials for
Functional Analysis II

winter term 2011/2012 (WiSe 2011/2012)

weekly schedule:

 Monday 2-4 pm room C-113 lecture Tuesday 10-12 am room C-112 lecture Wednesday 2-4 pm room B-132 tutorial session Wednesday 6-8 pm room C-111 excercise session

office hours:

 T. Ø. Sørensen Thursday 10-11 am office B-408 A. Michelangeli Wednesday 5-6 pm office B-334

final written test:
• The final written test will take place on Saturday 4 February at 9 am in room B-132

FINAL TEST            solutions

homework assignments:
• Each homework assignment is posted by Wednesday evening.
• Hand-in deadline: each Wednesday by 2 pm, in the designated drop-box located on the first floor.
• Pick up your marked worksheets from the designated return box on the first floor.
• Markers: Sebastian Gottwald, Valentina Ros (contact them for clarifications on marking).
• A sketch of the solution to each exercise will be posted weekly.
• A selection of the homework exercises will be discussed in detail in the homework session.
• Please register through the LMU Maths Institute homework webpage.

 week 1 assignment 01 solutions problems in class, set 01 solutions week 2 assignment 02 solutions problems in class, set 02 solutions week 3 assignment 03 solutions problems in class, set 03 solutions week 4 assignment 04 solutions problems in class, set 04 solutions week 5 assignment 05 solutions problems in class, set 05 solutions week 6 assignment 06 solutions problems in class, set 06 solutions week 7 assignment 07 solutions problems in class, set 07 solutions week 8 assignment 08 solutions problems in class, set 08 solutions week 9 assignment 09 solutions problems in class, set 09 solutions week 10 assignment 10 solutions problems in class, set 10 solutions week 11 assignment 11 solutions problems in class, set 11 solutions week 12 assignment 12 solutions problems in class, set 12 solutions week 13 assignment 13 solutions problems in class, set 13 solutions

 further topics in the tutorials (besides all problems in class): warm-up overview from FA                                                        completeness Banach vs Hilbert adjoint approximations of C[0,1] via Bernstein polynomials                                                         The Lax-Milgram theorem "Question time" 7 Dec 2011               X-MAS quiz closed operators integration by parts in AC[0,1]

(E=Exercise, P=Problem in class)

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week 13
assignment 13   problems in class, set 13

E49   Free energy on [0,1] with Dirichlet/Neumann boundary conditions. Essential self-adjointness.
E50   All self-adj extension of the momentum on [0,1] defined with Dirichlet boundary conditions.
E51   Essential self-adjointness is not preserved in the strong operator limit.sd
E52   Trotter: convergence of the unitary groups implies norm resolvent convergence.

P49   Momentum operator on [0,1] with Dirichlet boundary conditions. Closure, adjoint. Spectrum changes!
P50   The spectrum of non closed operators is the whole C-plane.
P51   Canonical commutation relations cannot be satisfied by bounded operators.
P52   Stone’s theorem.

further topics in the tutorial: Klausur warm-up

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week 12
assignment 12   problems in class, set 12

E45   A concrete bdd self-adj operator reduced in multiplication form.
E46   Example of ops having or not cyclic vectors. Cyclic vectors for the position operator don't vanish a.e.
E47   Unbounded multiplication operator. Unbounded position operator.
E48   A symmetric op with an orhtonormal basis in its domain is essentially self-adjoint.

P45   (aA)*, (A+B)*, (AB)*. If B extends A, A* extends B*. Self-adjoint operators are maximal.
P46   Adjoint of the inverse, adjoint of the closure. If A is injective and self-adjoint, so is its inverse.
P47   The domain of the adjoint can be quite small.
P48   von Neumann’s theorem for conjugations. Standard Schroedinger operators are essentially self-adj.

further topics in the tutorial: from bounded to unbounded operator - outlook

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week 11
assignment 11   problems in class, set 11

E41   Operator convex functions.
E42   Spectral resolution of the position operator.
E43   The restriction of a self-adjoint operator to a spectral subspace.
E44   Applications of the spectral theorem: unitary group exp(itA), norm of the resolvent.

P41   Fuglede's thm (need functional calc. for normal ops): if [S,T]=O, T is normal, then [S,T*]=O.
P42   A hermitian matrix has cyclic vectors iff all its eigenvalues are distinct.
P43   0<A<1 iff A^2<A. Proof with and without spectral theorem (multiplication form).
P44   Riesz projections.

further topics in the tutorial: retrospective outlook on spectral theorem

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week 10
assignment 10   problems in class, set 10

E37   Positive operators and their invertibility.
E38   |A| with and without the functional calculus. Decomposition A=(A+)-(A-).
E39   Operator monotone functions. A<B => A^a < B^a for a in [0,1], not for a>1.
E40   Application of the spectral mapping theorem. Stone's formula.

P37   Functional calculus for commuting self-adjoint ops. Functional calculus for normal operators.
P38   No spectral mapping thm if the operator is not normal.
P39   Properties of a p.v.m.
P40   Well-posedness of the definition of integral with respect to a p.v.m.

further topics in the tutorial: X-mas quiz!

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week 9
assignment 09   problems in class, set 09

E33   Spectrum of a partial isometry.
E34   Examples of polar decompositions.
E35   The discrete Laplacian: norm and spectrum.
E36   The spectrum of operator-norm upper semicontinuous, but not continuous.

P33   The spectral radius is semi-continuous but not norm continuous.
P34   The spectrum is norm continuous for normal operators.
P35   Position operator on [0,1]: norm, self-adjointness, no eigevalues, spectrum=[0,1].
P36   Multiplication operator on a finite measure space: norm, adjoint, spectrum=essential range.

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week 8
assignment 08   problems in class, set 08

E29   Min-Max for bounded self-adjoint operator.
E30   Operators with spectral radius r<1: they induce an equivalent norm.
E31   The spectral radius is invariant under similarity. Variational caracterisation.
E32   Check of continuous/compact embeddings between various functional spaces.

P29   A is positive iff spec(A) lies on the positive semi-axis.
P30   T (and T*) is invertible iff T and T* are bounded away from zero. Iff T and T* are injective with close range.
P31   Polar decomposition of invertible operators (the partial isometry is unitary).
P32   Russo-Dye thorem: if ||T||<1 then T is a mean of unitaries. Fails to hold if ||T||=1.

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week 7
assignment 07   problems in class, set 07

E25   Normal operators and their invertibility. The shift cannot be the finite product of normal operators.
E26   An operator that acts the same way on different Banach spaces and has different spectra.
E27   Spectral radius of a product and of a sum.
E28   Variational characterisation of the norm of a selfadj operator.

P25   Relations between Ran and Ker of T and T*
P26   ||T|| and sup|<x,Tx>| (||x||=1) are equivalent norms.
P27   Decomposition of any self-adjoint in two unitaries and of any bounded operator in four uniaries.
P28   Weyl criterion for normal operators. An isolated point in the spectrum of a normal op. is an eigenvalue.

further topics in the tutorial: question-time

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week 6
assignment 06   problems in class, set 06

E21   Hardy operator on C[0,1]: bounded, no compact, eigevnalues.
E22   Hardy operator on C'[0,1]: bounded, compact, eigenvalues.
E23   Hardy operator on L^p[0,1]: bounded, no compact, compute norm.
E24   Hardy operator on L^p[0,inf]: bounded, no compact, compute norm. Hardy's inequality.

P21   Positive bounded operators on a complex Hilbert space are self-adjoint
P22   Square root of a positive bounded operator (not necessarily compact).
P23   Counterexamples to |A+B|<|A|+|B| for operators. Schwarz ineq for AB (operators)
P24   Unitary operators vs Isometries vs Partial Isometries.

further topics in the tutorial: Lax-Milgram

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week 5
assignment 05   problems in class, set 05

E17   Compact and continuous injection. (Lyon's lemma.)
E18   Weyl sequences, Weyl criterion: general case, self-adj, unitary. Given a compact spectrum, find the operator.
E19   Compact operators with zero spectrum.
E20   Idempotent operators and spectrum.

P17   Solving an integral equations of the form Tf=af for T compact.
P18   Spectral radius formula: lim version and inf version.
P19   Examples of r(T) strictly smaller than ||T||
P20   Decomposition of a bounded operator into real and imaginary part. Special case for normal and unitari ops.

further topics in the tutorial: approximation of C[0,1] by Bernstein polynomials

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week 4
assignment 04   problems in class, set 04

E13   Volterra integral operator on H-space: compute the resolvent
E14   Perturbation of the spectrum with compact operators
E15   Canonical form of compact operators on a Hilbert space
E16   Compute an explicit decomposition of a compact operator.

P13   Spectrum of the product
P14   Compact operators on reflexive Banach spaces (weak-to-norm convergence)
P15   dim(ker(1-T)) where T is an integral operator
P16   Spectrum of the polynomial of an operator

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week 3
assignment 03   problems in class, set 03

E09   boundedness, possible compactness, adjoint, spectrum of multiplicaction on l2
E10   Volterra integral operator on H-space: compactness, non-zero ev, spectrum
E11   Volterra integral operator on H-space: adjoint, Riesz-Schauder decomposition, norm
E12   Resolvent is continuous and uniformly holomorphic

P09   Integral operators on H-space. Checks of compactness.
P10   Spectrum of T^{-1}
P11   Projections on vector / normed / Banach space (and proof of Remark 1.2 from class)

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week 2
assignment 02   problems in class, set 02

E05   Compact operators: 0 is in the closure of T(unit sphere)
E06   Orthogonal projections on H-space. Spectrum. Resolvent.
E07   Compact operators: only finite dimensional nonzero eigenvalues

P05   Operators of the form 1-T, T compact
P07   Resolvent: useful identities and inequalities
P08   Point/cont/residual spectrum of an explicit T and of T'

further topics in the tutorial: spectral types, adjoints,
why we care so much about compactness

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week 1
assignment 01   problems in class, set 01

E01   Compact operators: concrete check of compactness
E02   Compact operators: they attain their norm
E03   Compact operators: never surjective on infinite dim spaces
E04   Compact operators: if they are projections, they are finite rank

P01   Compact operators: concrete examples.
P02   Compact operators: ideal property, 0 in spec, weak to norm convergence
P03   Compact operators: non-injective compact ops are dense in the compacts
P04   Compact operators: norm-limit of compact is compact. Approximation with finite rank.

further topics in the tutorial: review from FA1, completeness

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