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

diary of homework assignments and
tutorials (table of contents):
(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 selfadjointness.
E50 All selfadj extension of the momentum on [0,1] defined with
Dirichlet boundary conditions.
E51 Essential selfadjointness 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 Cplane.
P51 Canonical commutation relations cannot be satisfied by
bounded operators.
P52 Stone’s theorem.
further topics in the tutorial: Klausur warmup
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week 12
assignment 12
problems in
class, set 12
E45 A concrete bdd selfadj 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 selfadjoint.
P45 (aA)*, (A+B)*, (AB)*. If B extends A, A* extends B*.
Selfadjoint operators are maximal.
P46 Adjoint of the inverse, adjoint of the closure. If A is
injective and selfadjoint, 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 selfadj.
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 selfadjoint 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 selfadjoint 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 Wellposedness of the definition of integral with respect to
a p.v.m.
further topics in the tutorial: Xmas 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 operatornorm upper semicontinuous, but not
continuous.
P33 The spectral radius is semicontinuous but not norm
continuous.
P34 The spectrum is norm continuous for normal operators.
P35 Position operator on [0,1]: norm, selfadjointness, 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 MinMax for bounded selfadjoint 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 semiaxis.
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 RussoDye 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 selfadjoint 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:
questiontime
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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
selfadjoint
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: LaxMilgram
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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, selfadj,
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 Hspace: 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
(weaktonorm convergence)
P15 dim(ker(1T)) 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 Hspace: compactness, nonzero
ev, spectrum
E11 Volterra integral operator on Hspace: adjoint,
RieszSchauder decomposition, norm
E12 Resolvent is continuous and uniformly holomorphic
P09 Integral operators on Hspace. Checks of compactness.
P10 Spectrum of T^{1}
P11 Projections on vector / normed / Banach space (and proof of
Remark 1.2 from class)
P12 Spectrum of selfadjoint operators
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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 Hspace. Spectrum. Resolvent.
E07 Compact operators: only finite dimensional nonzero
eigenvalues
E08 Adjoint: compute an explicit (easy) adjoint L^3 to L^2
P05 Operators of the form 1T, T compact
P06 Adjoint: compute explicit adjoint (GinibreVelo's TT* for
Strichartz inequality)
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: noninjective compact ops are dense
in the compacts
P04 Compact operators: normlimit of compact is compact.
Approximation with finite rank.
further topics in the tutorial: review from FA1, completeness
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