• Keine Ergebnisse gefunden

Functional Analysis II – Problem sheet 2

N/A
N/A
Protected

Academic year: 2022

Aktie "Functional Analysis II – Problem sheet 2"

Copied!
2
0
0

Wird geladen.... (Jetzt Volltext ansehen)

Volltext

(1)

Functional Analysis II – Problem sheet 2

Mathematisches Institut der LMU Prof. Dr. P. M¨uller, Dr. A. Michelangeli

Handout: 6.05.2009

Due: Tuesday 12.05.2009 by 10,15 a.m. in the “Funktionalanalysis II” box Questions and infos: Dr. A. Michelangeli, office B-334, michel@math.lmu.de Grader: Ms. S. Sonner – ¨Ubungen on Wednesdays, 4,30 - 6 p.m., room C-111

Exercise 4. Prove that any bounded measurable function onRcan be uniformly approximated by a step function that has finitely many levels.1 In other words, pick a bounded measurable function f : R R (for a generic f : R C one just repeats the argument for the real and imaginary part of f) and fix an arbitrarily small error ε > 0: then prove that there exists a step function fε : R C, with finitely many levels, such that kf −fεk 6 ε. (Hint: the natural way to proceed is toconstruct explicitly the approximating step function. To this aim, observe that the range of f can be covered by a finite and conveniently large number of balls.

Consider the pre-image of these balls to define the steps of fε. This also shows you that the number of levels depend on ε.)

Exercise 5. (The discrete Laplacian on Zd.) Recall that the space

`2(Zd) := ©

φ :Zd C : X

x∈Zd

|φ(x)|2 <∞ª is a Hilbert space with the scalar product hφ, ξi = P

x∈Zdφ(x)ξ(x). You may think of an element φ in `2(Zd) just as an assignment of complex numbers, one at each point of the infinite lattice Zd, such that they are square summable. (You should be familiar with the one-dimensional version of it, namely `2 =`2(Z).) The goal of this problem is to introduce a self-adjoint operator on`2(Zd) which is the discrete analogue of the usual Laplacian onL2(Rd), with the key difference that the discrete version is bounded. As a consequence, by means of the spectral theory for bounded operators that you are supposed to know until know, you should be able to determine its spectrum.

Recall first the following basic facts (if you did not know them already, you are strongly encouraged to prove them separately, although this is not part of the exercise). The operator R :`2(Z)→`2(Z) such that (Rφ)(x) = φ(x−1) for allx∈Z is called the right shift operator. R is unitary and its adjoint L :=R : `2(Z) →`2(Z) is just the left shift operator, that is, (Lφ)(x) =φ(x+ 1) for all x∈Z. Moreover Spec(R) = Spec(L) ={λ∈ C : |λ|= 1}.

1With this nomenclature one means, for instance, that theRRfunction

g(x) = (

1 x[2k,2k+ 1)

0 x[2k+ 1,2k+ 2) (kZ)

is astep function with of course two “levels” only (0 and 1), but with an infinite number of “steps”.

1

(2)

Here the problem starts.

5.1) By definition the discrete Laplacian is the operator ∆ : `2(Zd) `2(Zd) acting on any element φ ∈`2(Zd) as

(∆φ)(x) := X

yZd

|x−y|=1

¡φ(x)−φ(y)¢

, x∈Zd. (∗)

In the notation above |x −y| is the Euclidean distance between x and y as points of Rd with integer coordinates. How many terms are there in the r.h.s. of (∗)? Rearrange the summands in the r.h.s. of (∗) so to express the operator ∆ in terms of the identity operator and a number of shift operators.

5.2) Prove that ∆ is bounded with norm k∆k= 4d. Prove also that ∆ = ∆. (Hint: one can certainly prove both statements from the scratch and this would be fully graded as well, but it is a pain for you! (and for the grader too.) Alternatively, note that, at least for proving self-adjointness and k∆k6 4d, the rearrangement in point (5.1) above does the job.)

5.3) Prove the operator inequality06∆64d.

5.4) Prove that Spec(∆)[0,4d].

5.5) Prove that actually Spec(∆) = [0,4d]. (Hint: if you have completed the previous point, you are left with proving that Spec(∆) [0,4d]. A possible way is to take any λ∈[0,4d]

and to prove that λsatisfies the Weyl’s criterion (→Exercise 3). To this aim, you need to identify one Weyl sequence, namely a sequencen}n=1 ⊂`2(Zd) such thatk(∆−λ)φnk → 0 asn → ∞. Here is a suggestion to constructφnexplicitly. Consider theZdCfunction

e

ϕk(x) = eik·x in the variable x = (x1, . . . , xd) Zd, where k = (k1, . . . , kd) Rd is fixed and k · x = Pd

j=1kjxj is the Euclidean scalar product of k times x as points of Rd. Observe thatϕek∈/`2(Zd), nevertheless compute theformal action of ∆ on ϕek, i.e., do the computation just by means of the prescription (∗). This way you should see that given any λ [0,4d] you can always choose k depending on λ such that “∆ϕek =λϕek”. Hence the “eigenvalue” λ is in the spectrum of ∆. Of course this is formal becauseϕek does not belong to the domain of ∆ but it should give you a hint on how to construct the Weyl sequence n}n=1 you are looking for. More concretely, what is the difference if in the formal argument above you modify ϕek setting it to give zero for allx∈Zdoutside a large cube centred at the origin?)

2

Referenzen

ÄHNLICHE DOKUMENTE

a certain graph, is shown, and he wants to understand what it means — this corre- sponds to reception, though it involves the understanding of a non-linguistic sign;

1) Inventory, reprocessing and analysis of existing data from the Weddell Sea to evaluate and identify key areas, which are in the need of protection. 2) Discussion, improvement

To match the market stochasticity we introduce the new market-based price probability measure entirely determined by probabilities of random market time-series of the

The structure which had been proposed for the pyoverdin named pseudobactin 7SR1 (Yang and Leong, 1984) differed from those of all other pyoverdins investigated sofar: its peptide

Solutions based on trusted devices can be classified into standalone approaches, where the trusted device (TD) is not attached to the user’s untrusted platform, and connected

The aim of this paper is to compare the forecast performance of three structural econometric models 1 ; the ARIMAX model, the Kalman filter model and the non- parametric model

The red-green government of Chancellor Gerhard Schröder enforced promotion of electricity produced from renewable energy sources and the gradual restriction of

The Machinery Directive harmonises the requirements for machinery and for products that are equated to machinery (e.g. interchangeable equipment, safety components) and to