Problem Set 10

Humboldt-Universit¨at zu Berlin Institut f¨ur Mathematik

C. Wendl, S. Dwivedi, L. Upmeier zu Belzen

Funktionalanalysis

WiSe 2020–21

Problem Set 10

Due: Thursday, 11.02.2021 (15pts + 5 for free)¹

Problems marked with p˚q will be graded. Solutions may be written up in German or English and should be submitted electronically via the moodle before the ¨Ubung on the due date. For problems withoutp˚q, you do not need to write up your solutions, but it is highly recommended that you think through them before the next Tuesday lecture. You may also use the results of those problems in your written solutions to the graded problems.

Problem 1

For this problem, we consider functions valued in a fixed finite-dimensional complex inner product space pV,x , yq. Recall that for s P R, the Hilbert space H^spRⁿq is defined to consist of all tempered distributions f P S¹pRⁿq whose Fourier transforms fpP S¹pRⁿq are represented by functions of the form fpppq “ p1` |p|²q^´s{2gppq for some g P L²pRⁿq.

The inner product onH^spRⁿq is given by xf, gy_H^s :“

A

p1` |p|<sup>2</sup>q<sup>s{2</sup>f ,pp1` |p|²q^s{2pg E

L².

If a distributionf PH^spRⁿqis representable by a locally integrable function, we generally identify it with this function; note that this is always possible whensě0, but not when să0. For an open subset ΩĂRⁿ, the closure in H^spRⁿq of the spaceC₀⁸pΩq of smooth functions on Rⁿ with compact support in Ω defines a closed subspace Hr^spΩq ĂH^spRⁿq, and the quotient of H^spRⁿq by the closed subspace of distributions that vanish on test functions supported in Ω is denoted byH^spΩq.

(a) p˚q GivennPN, for whichsPRis the Diracδ-distribution in H^spRⁿq? [3pts]

(b) Prove that SpRⁿq is dense inH^spRⁿq for everysPR.

(c) Prove that the pairing SpRⁿq ˆSpRⁿq Ñ C : pϕ, ψq ÞÑ xϕ, ψy_L² extends to a continuous real-bilinear pairing

H^´spRⁿq ˆH^spRⁿq ÑC:pf, gq ÞÑ xf, gy:“ xp1` |p|<sup>2</sup>q<sup>´s{2</sup>f ,pp1` |p|²q^s{2pgy_L², such that the real-linear map f ÞÑ xf,¨y sends H^´spRⁿq isomorphically to the dual space of H^spRⁿq.

(d) Given an open subset Ω with compact closure inp0,1qⁿ, associate to eachf PC₀⁸pΩq the unique function F PC⁸pTⁿq such that fpxq “Fpxq for x P p0,1qⁿ. Show that the map C₀⁸pΩq ÑC⁸pTⁿq:f ÞÑF extends to bounded linear injections

L²pΩq –Hr⁰pΩqãÑL²pTⁿq and Hr¹pΩqãÑH¹pTⁿq whose images are closed.

Hint: Avoid Fourier analysis here by replacing the usualH¹-norm with the equivalent norm }u}:“ř

|α|ď1}B^αu}_L². This works equally well onRⁿ orTⁿ.

1This version of the problem set has been revised to correct some errors that invalidated the original version of Problem 1(j) (worth 5 points).

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Problem Set 10

(e) Deduce from the compactness of the inclusion H¹pTⁿq ãÑ L²pTⁿq that the map Hr^spΩq ÑH^´spΩq:f ÞÑ rfsis compact for every sě1 and every bounded open set ΩĂRⁿ.

(f) Let ∆ :“ř_n

j“1B²_j denote the Laplace operator. Show that the linear map Φ :SpRⁿq ÑSpRⁿq:uÞÑu´ 1

4π²∆u

has a unique extension to a unitary isomorphism Φ :H¹pRⁿq ÑH^´1pRⁿq.

(g) Let I : H^´1pRⁿq Ñ pH¹pRⁿqq^˚ denote the real-linear isomorphism from part (c).

Show that the mapHr¹pΩq Ñ`

Hr¹pΩq˘˚

:uÞÑIΦpuqˇ ˇ

Hr¹pΩqis an isometric real-linear isomorphism, and deduce that Hr¹pΩq ÑH^´1pΩq:uÞÑ rΦpuqsis an isomorphism.

Hint: Write down an explicit formula for IΦpuqf foru, f PHr¹pΩq.

(h) Deduce that Hr¹pΩq ÑH^´1pΩq:uÞÑ r∆us is a Fredholm operator of index 0.

(i) Show that the equation ∆u“0 has no nontrivial solutionsuPC₀⁸pRⁿq.

Hint: What does integration by parts tell you about ş

Rⁿxu,∆uydm?

(j) Prove thatHr¹pΩq ÑH^´1pΩq:uÞÑ r∆usis an isomorphism.

Hint: Extend the formula for ş

Rⁿxu,∆uydm in part (i) to all u P Hr¹pΩq, and use this to prove injectivity.

Problem 2

AssumeXis a complex Banach space andT PLpXq. We say thatλPCis anapproximate eigenvalue of T if there exists a sequence xn P X with }xn} “ 1 for all n such that pλ´Tqx_nÑ0. Prove:

(a) Every approximate eigenvalue of T belongs to the spectrum σpTq.

(b) p˚q If λP σpTq is neither an eigenvalue nor belongs to the residual spectrum of T, then it is an approximate eigenvalue of T. [4pts]

(c) p˚q For the operator T : `<sup>1</sup> Ñ `¹ : px₁, x₂, x₃, . . .q ÞÑ px₂, x₃, x₄, . . .q, 1 is not an eigenvalue but is an approximate eigenvalue. [4pts]

Problem 3

Given a complex Banach spaceX and T PLpXq, let T¹ PLpX^˚q denote the transpose, also known as the dual operator ofT.² Prove:

(a) If λPσpTq is in the residual spectrum of T then it is an eigenvalue of T¹.

(b) p˚qIfλPσpT¹q is an eigenvalue ofT¹, then it is either an eigenvalue ofT or belongs to the residual spectrum of T. [4pts]

Now suppose X is a complex Hilbert space H, and T^˚ : H Ñ H denotes the adjoint operator, defined via the conditionxx, T yy “ xT^˚x, yyfor all x, yPH. Prove:

(c) σpT^˚q “ λsPC ˇ

ˇλPσpT¹q( (d) σpTq “σpT¹q

Hint: T^˚ : H Ñ H and T¹ : H^˚ Ñ H^˚ are closely related via the complex-antilinear isomorphismHÑH^˚:xÞÑ xx,¨y.

2We have sometimes denotedT¹ in the past byT^˚, but will now be reserving the latter notation for the adjoint of an operator on a complex Hilbert space.

2