18.155, PROBLEM SET 5Problem 5.1. Let A be a self-adjoint (meaning A∗= A) compact operator on aseparable Hilbert space H. Recall from class that an eigenvalue of A is a complexnumber λ such that A − λ Id has non-trivial null space and that the eigenvalues ofA (whether self-adjoint or not) form a discrete subset of C \ {0} and that for eachλ the space of associated generalized eigenvectors is finite dimensional.(1) Show that any eigenvalue of A is real.(2) Show that every generalized eigenvector, that is a solution of (A−λ Id)ku =0 for some k and λ 6= 0, is actually an eigenvector. Hint:- Show that A actson the generalized eigenspace Eλcorresponding to λ and is a self-adjointmatrix and then apply your knowledge of self-adjoint matrices.(3) Show that the non-zero eigenvalues of A2are positive and that t2> 0 is aneigenvalue of A2if and only if either t or −t is an eigenvalue of A and thatthe eigenspace of t2is the sum of the eigenspaces of A with eigenvalues ±t(where the eigenspace of s is interpreted as {0} if s is not an eigenvalue).(4) Show that if A is not identically zero then A has an eigenvalue. Hint:- Lookat the space of u ∈ H with kuk = 1 such that kA2uk2= kA2k. Then choosea sequence unwith kunk = 1 and kA2unk → kA2k. Show that unhas aweakly convergent subsequence such that Aunkconverges and check thatthe limit is in the desired space. Conclude that A has a non-zero eigenvalue.(5) Prove that the space N⊥, the orthocomplement in H of the null space ofA, has an orthonormal basis of eigenvectors of A.Problem 5.2. Let A be a self-adjoint Hilbert-Schmidt operator on a separableHilbert space H. Using the results of the previous problem, show that the non-zeroeigenvalues λjof A, repeated with the multiplicity (dimension of the associatedeigenspace), are such that(1)Xjλ2j< ∞.Problem 5.3. Let T be a self-adjoint operator of trace class on a separable Hilbertspace. Show that the eigenvalues, repeated with their multiplicities, satisfy(2)Xj|λj| < ∞and that(3) Tr(T) =Xjλj.Problem 5.4. Consider the operator on L2(Rn), depending on a parameter s > 0,(4) As: L2(Rn) 3 u −→ F−1(1 + |ξ|2)−s/2ˆu ∈ L2(Rn).12 PROBLEMS 5(1) Show that if s > n/2 then Ascan be written in the form(5) Asu(x) =ZRnKs(x − y)u(y)dy, Ks(z) ∈ L2(Rn).(2) Show that, again for s > n/2, the ope rator on L2(B), with B the unit ballin Rb, given by(6) Gsu = χ(As(χu)),where χ is the characteristic function of B, is Hilbert-Schmidt.Problem 5.5. Recall from class the operator A which solves the Dirichlet problemin a bounded open domain Ω ⊂ Rn. It is the case that A is compact and self-adjoint.Deduce that there is an orthonormal basis of L2(Ω) composed of eigenvec tors ofthe Dirichlet
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