1 Math 241A-B Homework Problem List for F2015 and W2016

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1 1 Math 241A-B Homework Problem List for F2015 W Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let f g := sup [g (x) f (x) Y ] x U 1.6 Homework 6. Due Friday, January 22, 2016 H in the following three exercises: 1.32, 1.33, Homework 7. Due Friday, February 19, 2016 B g (U, V ) := { } f : U Y f g <. H in the following three exercises: 1.37, 1.40, 1.41, 1.43, 1.45 Look at exercises: 1.36, 1.38, 1.39, 1.42, H in exercises: 1.1, 1.2 Look at exercises: Homework 2. Due Wednesday, October 14, 2015 H in exercises: 1.4, 1.8, 1.9 Look at from lecture note exercises: 1.5, 1.6, 1.7, 1.11, 1.12, Homework 8. Due Friday, Marh 4, 2016 H in the following three exercises: 1.47, 1.48, 1.52, Look at exercises: 1.46, 1.49, 1.50, Homework 3. Due Wednesday, October 30, 2015 H in exercises: 1.10, 1.16 Look at exercises: 1.15, 1.17, 1.18, 1.19, Homework 4. Due Friday, November 13, 2015 H in exercises: 1.22, 1.23, 1.24 Look at exercises: Homework 5. Due Friday, December 5, 2015 H in the following three exercises: 1.26, 1.28 Look at exercises: 1.25, 1.27, 1.29, 1.30, 1.31.

2 2 1 Math 241A-B Homework Problem List for F2015 W2016 Exercise 1.1. Let Y = BC (R, C) be the Banach space of continuous bounded complex valued functions on R equipped with the uniform norm, f u := sup x R f (x). Further let C 0 (R, C) denote those f C (R, C) such that vanish at infinity, i.e. lim x ± f (x) = 0. Also let C c (R, C) denote those f C (R, C) with compact support, i.e. there exists N < such that f (x) = 0 if x N. Show C 0 (R, C) is a closed subspace of Y that C c (R, C) = C 0 (R, C). Exercise 1.2. Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let let f g := sup [g (x) f (x) Y ] x U B g (U, V ) := { } f : U Y f g <. Verify that g makes B g (U, Y ) into a normed space. Moreover show; if Y ) is a Banach space, then (B g (U, Y ), g is a Banach space. Exercise 1.3. Now assume that U is a subset of R n (or more generally any topological space) g C (U, (0, )). Show B g C (U, Y ) := B g (U, Y ) C (U, Y ) is a closed subspace of B g (U, Y ). In particular if Y is a Banach space, then B g C (U, Y ) is a also a Banach space. Exercise 1.4 (Prove the BLT Theorem). Suppose that Z is a normed space, X is a Banach space, S Z is a dense linear subspace of Z. If T : S X is a bounded linear transformation (i.e. there exists C < such that T z C z for all z S), then T has a unique extension to an element T L(Z, X) this extension still satisfies T z C z for all z S. Exercise 1.5. Assume the norms on X Y are the l 1 norms, i.e. for x R n, x = n j=1 x j. Then the operator norm of T is given by T = max 1 j n i=1 m T ij. Exercise 1.6. Assume the norms on X Y are the l norms, i.e. for x R n, x = max 1 j n x j. Then the operator norm of T is given by T = max 1 i m j=1 n T ij. Exercise 1.7. Assume the norms on X Y are the l 2 norms, i.e. for x R n, x 2 = n j=1 x2 j. Show T 2 is the largest eigenvalue of the matrix T tr T : R n R n. Hint: Use the spectral theorem for symmetric real matrices. Exercise 1.8. Suppose that X is a normed space there exists an uncountable set A {x t } t A X such that x t x s 1 for all s t with s, t A. Show X is not separable. Exercise 1.9. Let X = B ( L 2 (R, dx), L 2 (R, dx) ) be the Banach space of bounded operators from L 2 (R, dx) to itself. Show X is not separable by showing; T t T s op 2 1 for s t where for t R, T t X is define by As usual, (T t f) (x) = f (x + t) for x R f L 2 (R, dx). T op := sup T f L2 (dx) f L 2 (dx) =1 f 2 L 2 (dx) = R f (x) 2 dx. Exercise 1.10 ({T t } t R is strongly continuous). Let X = B ( L 2 (R, dx), L 2 (R, dx) ) be the Banach space of bounded operators from L 2 (R, dx) T t X is define by (T t f) (x) = f (x + t) for x R f L 2 (R, dx). Show lim t 0 T t f f L 2 (dx) = 0 for all f L2 (dx). Hint: make use of the fact that C c (R) is dense in L 2 (dx). Exercise Let X = N for p, q [1, ) let p denote the l p (N) norm. Show p q are inequivalent norms for p q by showing f p sup = if p < q. f 0 f q Exercise Let d : C(R) C(R) [0, ) be defined by d(f, g) = n=1 2 n f g n 1 + f g n, where f n := sup{ f (x) : x n} = max{ f (x) : x n}. Page: 2 job: 241ahmsolns macro: svmonob.cls date/time: 5-Feb-2016/11:29

3 1.8 Homework 8. Due Friday, Marh 4, Show that d is a metric on C(R). 2. Show that a sequence {f n } n=1 C(R) converges to f C(R) as n iff f n converges to f uniformly on bounded subsets of R. 3. Show that (C(R), d) is a complete metric space. Exercise Let X = C([0, 1], R) for f X, let f 1 := 1 0 f(t) dt. Show that (X, 1 ) is normed space show by example that this space is not complete. Hint: For the last assertion find a sequence of {f n } n=1 X which is trying to converge to the function f = 1 [ 1 2,1] / X. Exercise Suppose X is a Banach space {f n : n N} is a sequence in X such that lim n f n = f X. Show s N := 1 N N n=1 f n for N N is also a convergent sequence lim N 1 N N n=1 f n = lim N s N = f. Exercise Let (H, ) be a Hilbert space suppose that {P n } n=1 is a sequence of orthogonal projection operators on H such that P n (H) P n+1 (H) for all n. Let M := n=1p n (H) (a subspace of H) let P denote orthonormal projection onto M. Show lim n P n x = P x for all x H. Hint: first prove the result for x M, then for x M then for x M. Exercise 1.16 (A Martingale Convergence Theorem). Suppose that {M n } n=1 is an increasing sequence of closed subspaces of a Hilbert space, H, M := n=1 M n, P n := P Mn, {x n } n=1 is a sequence of elements from H such that x n = P n x n+1 for all n N. [We call such a sequence a martingale.] Show; 1. P m x n = x m for all 1 m n <, 2. (x n x m ) M m for all n m, 3. x n is increasing as n increases, 4. if sup n x n = lim n x n <, then x := lim n x n exists in M that x n = P n x for all n N. (Hint: show {x n } n=1 is a Cauchy sequence.) Exercise 1.18 (Riemann Lebesgue Lemma for Fourier Series). Show for f L 1 ([ π, π] d) that f c 0 (Z d ), i.e. f : Z d C lim k f(k) = 0. Hint: If f L 2 ([ π, π] d), this follows from Bessel s inequality. Now use a density argument. Exercise Let f L 1 (( π, π]) which we extend to a 2π periodic function ( on ) R continue to denote by f. If there exists q N such that f x + 2π q = f (x) for m a.e. x, then f (k) = 0 unless q divides k. Exercise In this problem we assume the notation from subsection?? with d = 1. For simplicity of notation we identify L 2 (( π, π], dθ) with 2π periodic functions on R via, L 2 (( π, π], dθ) f n Z f (x + n2π) 1 ( π,π] (x + n2π) L 2 per (R). Given α R let (U α f) (θ) = f (θ + α2π) wherein we have used the above identification. If α / Q show M α = Nul (U α I) = C 1. If α Q write α = p q where gcd (q, p) = 1, i.e. p q are relatively prime. In this case show M α = Nul (U α I) consists of those f L 2 per (R) such that ( ) f x + 2π q = f (x) for m a.e. x. [Consequently, combining this exercise with Mean Ergodic Theorem?? shows, n 1 1 n k=0 U k α s P Mα where M α depends on α as described above.] Exercise Let L = α k a α (x) α (1.1) with a α P. Show L(S) S in particular α f x α f are back in S for all multi-indices α. Exercise Let H K be Hilbert spaces S H T K be total sets. If there exists a surjective map, f : S T, such that f (s), f (s ) K = s, s H for all s, s S, then there exists a unique unitary map, U : H K, such that Us = f (s) for all s S. (a multi- Exercise Recall that for f C (R n ), N N 0, α N n 0 index) we let f N,α = sup (1 + x ) N α f (x), x R n where Page: 3 job: 241ahmsolns macro: svmonob.cls date/time: 5-Feb-2016/11:29

4 4 1 Math 241A-B Homework Problem List for F2015 W2016 n ( ) αi α := α αn n =. x i=1 i With this notation, the Schwartz test function space is defined to be { } S = S (R n ) = f C (R n ) : f N,α < for all N α. Show S is complete, i.e. show if {f n } n=1 S satisfies lim f n f m m,n N,α = 0 N N 0 α N n 0, then there exists f S such that lim f f n n N,α = 0 N N α N n 0. Exercise Let A = b 1... b k where b 1,..., b k { a, a }, A = b k... b 1, l = q p where q := # { j : b j = a }, p = # {j : b j = a}. Show; 1. there is a function, c A : N 0 [0, ) such that, for all n N 0, AΩ n = c A (n) Ω n+l (1.2) n (n 1)... (n k + 1) ca (n) (n + 1) (n + 2)... (n + k). (1.3) 2. A Ω n = c A (n l) Ω n l for all n N 0, i.e. c A (n) = c A (n l) where by convention, c A (k) = 0 if k < For all f S, ( ) ( ) ( ) Af N + 1 N N + k f ( ) k/2 N + k f. Exercise Show for f S (R) that; 1. For all x R, f (x) 1 ˆf = 1 2π 1 2π f (x) 1 2 [ R R ˆf (k) dk ˆf (k) 2 ( 1 + k 2) 1/2 dk]. 2. Use the last displayed inequality the basic properties of the Fourier transform to prove the Sobolev inequality, f (x) 2 1 [ ] f f 2 2 for all x R, where f 2 2 := R f (x) 2 dx. Exercise Suppose that U is an open subset of R or C F : U Ω X is a measurable function such that; 1. U z F (z, ω) is (complex) differentiable for all ω Ω. 2. F (z, ) L 1 (µ : X) for all z U. 3. There exists G L 1 (µ : R) such that Show is differentiable d dz F (z, ω) z Ω G (ω) for all (z, ω) U Ω. U z Ω F (z, ω) dµ (ω) = F (z, ω) dµ (ω) X Ω F (z, ω) dµ (ω). z Exercise 1.26 (L. Gårding s trick I.). Prove Theorem??, i.e. suppose that T t L(X) for t 0 satisfies; 1. (Semi-group property.) T 0 = Id X T t T s = T t+s for all s, t (Norm Continuity at 0+) t T t is continuous at 0, i.e. T t I L(X) 0 as t 0. Then show there exists A L(X) such that T t = e ta where e ta is defined in Eq. (??). Here is an outline of a possible proof based on L. Gårding s trick. 1. Using the right continuity at 0 the semi-group property for T t, show there are constants M C such that T t L(X) MC t for all t > Show t [0, ) T t L(X) is continuous. 3. For ε > 0, let S ε := 1 ε ε 0 T τ dτ L(X). Show S ε I as ε 0 conclude from this that S ε is invertible when ε > 0 is sufficiently small. For the remainder of the proof fix such a small ε > 0. Page: 4 job: 241ahmsolns macro: svmonob.cls date/time: 5-Feb-2016/11:29

5 1.8 Homework 8. Due Friday, Marh 4, Show T t S ε = 1 ε t+ε t T τ dτ = S ε T t conclude using the fundamental theorem of calculus that d dt T ts ε = 1 ε [T t+ε T t ] for t > 0 ( ) d dt Tt I 0+T t S ε := lim S ε = 1 t 0 t ε [T ε I]. 5. Using the fact that S ε is invertible, conclude A = lim t 0 t 1 (T t I) exists in L(X) that moreover, A = 1 ε (T ε I) S 1 ε d dt T t = AT t for t > Using step 5., show d dt e ta T t = 0 for all t > 0 therefore e ta T t = e 0A T 0 = I. Exercise Let H = l 2 S n (x 1, x 2,... ) = (x n+1, x n+2,... ). Show; 1. S n s 0 for all x l2 while S nx = x for all x l 2 therefore S n is not strongly convergent to 0. This shows the adjoint operation is strongly discontinuous. 2. Observe that S n w 0 weakly S n w 0 weakly, i.e. limn S n x y = 0 = lim n S nx y. On the other h verify that S n S n = I for all n from which it follows that the map (A, B) AB is not jointly continuous in the weak operator topology even though it is in the strong operator topology provided A is restricted to a bounded sets. Exercise Continuing then notation used in Example??, show σ ap (S) = S 1. Exercise Let X be a complex Banach space, T = (T 1,..., T n ) [B (X)] n, n σ ap (T) = λ Cn : inf (T j λ j I) x = 0 x =1. j=1 Exercise 1.27 (Duhamel s Principle I). Suppose that A : R L(X) is a continuous function V : R L(X) is the unique solution to the linear differential equation, Explain why σ ap (T) σ ap (T 1 ) σ ap (T 2 ) σ ap (T n ) (1.6) V (t) = A (t) V (t) with V (0) = I. Let x X h C(R, X) be given. Show that the unique solution to the differential equation: is given by ẏ (t) = A (t) y (t) + h (t) with y(0) = x (1.4) y (t) = V (t) x + V (t) Hint: compute d dt [V 1 (t) y (t)]. t 0 V (τ) 1 h(τ) dτ. (1.5) Exercise 1.28 (Conway, Exr. 4, p. 198). Suppose that X is a complex Banach space, D o C is a non-empty open set, f : D X is a function such that l f : D C is analytic for all l X, show f : D X is analytic, i.e. complex differentiable. Exercise 1.29 (Conway, Exr. 4, p. 198 cont.). Let H be a separable Hilbert space. Give an example of a discontinuous function, f : [0, ) H, such that t f (t), h is continuous for all t 0. then show σ ap (T) is compact by showing σ ap (T) is closed. Exercise Suppose that H is a separable Hilbert space A B (H) is non-negative. Show A 1 exists iff there exists ε > 0 so that A εi. Exercise Suppose that H is a separable Hilbert space T B (H) is a normal operator. Show T = T iff σ (T ) R. Exercise Let H be a separable Hilbert space A B (H) be nonnegative. T [B (H)] n. Show σ ap (T) = { λ C n : 0 σ ( (T λ) (T λ) )} where = { λ C n : (T λ) (T λ) is not invertible } (T λ) (T λ) := n (T j λ j ) (T j λ j ). j=1 Exercise Let A be a self-adjoint operator on an n dimensional Hilbert space (n < ) V. Show that the general spectral theorem of Theorem?? or Page: 5 job: 241ahmsolns macro: svmonob.cls date/time: 5-Feb-2016/11:29

6 6 1 Math 241A-B Homework Problem List for F2015 W2016 Theorem?? implies that A has an orthonormal basis of eigenvectors. Hints: you may assume from the outset that V = L 2 (Ω, F, µ) A = M f where (Ω, F, µ) is a finite measure space such that dim L 2 (µ) = n f : Ω R is a bounded measurable function. [A preliminary result you might want to first prove is; if dim L 2 (Ω, F, µ) = n, then there exists a partition Π = {Ω 1,..., Ω n } F of Ω so that µ (Ω i ) > 0 (for any A F) either µ (A Ω i ) = µ (Ω i ) or µ (A Ω i ) = 0 for 1 i n.] Exercise Suppose that T = (T 1, T 2,..., T n ) B (H) n is a collection of commuting bounded normal operators on a separable Hilbert space H. Show; if D B (H) is an operator such that [D, T j ] = 0 for all 1 j n, then [D, f (T)] = 0 for all bounded measurable functions, f : σ ap (T) C. [Note: by Theorem??, the assumption that [D, T j ] = 0 automatically implies [ ] D, Tj = 0.] Exercise Let H be a Hilbert space with O. N. basis e 1, e 2,.... Let θ j be a sequence of real numbers in (0, π/2). Let Let x j = (cos θ j )e 2j + (sin θ j )e 2j 1 j = 1, 2,... y j = (cos θ j )e 2j + (sin θ j )e 2j 1 j = 1, M 1 = closedspan {x j } j=1 M 2 = closedspan {y j } j=1. 1. Show that the closed span of M 1 M 2 (i.e., the closure of M 1 + M 2 ) is all of H. 2. Show that if θ j = 1/j then the vector z = j 1 e 2j 1 j=1 is not in M 1 + M 2, so that M 1 + M 2 H. Exercise Define f on [0, 1] by { 2 if x is rational f(x) = x if x is irrational. Find the spectrum of M f as an operator on L 2 (0, 1). Exercise Let H = l 2 (Z) = {a = {a j } j= : a 2 := Define U : H L 2 ( π, π) by (Ua)(θ) = 1 2π n= j= a n e inθ. a j 2 < }. It is well known that U is unitary (see Theorem??). For f in l 1 (Z) define (C f a) n = k= f(n k)a k. 1. Show that C f is a bounded operator on H that C f op f Find Cf explicitly show that C f is normal for any f in l 1 (Z). 3. Show that UC f U 1 is a multiplication operator. 4. Find the spectrum of C f, where { 1 if j = 1 f(j) =. 0 otherwise Exercise Find a bounded self-adjoint operator, A, with both of the following properties: 1. A has no eigenvectors, 2. σ(a) is set of Lebesgue measure zero in R. Hint 1: Such an operator is said to have singular continuous spectrum. Hint 2: Consider the Cantor set. See Rudin, 3rd Edition, Section Exercise Suppose that E is a projection valued measure on a measurable space, (R n, B = B R n) for some n N. If B B is an atom of E, then there exists a unique point λ B such that E (B) = E ({λ}). Exercise 1.43 (Decomposition by spectral type). Let A be a bounded Hermitian operator on a complex Hilbert space H. Suppose that A = λde(λ) is its spectral resolution. Denote by H ac the set of all vectors x in H such that the measure B µ x (B) := E(B)x 2 is absolutely continuous with respect to Lebesgue measure. 1. Show that H ac is a closed subspace of H. 2. Show that H p H ac. Page: 6 job: 241ahmsolns macro: svmonob.cls date/time: 5-Feb-2016/11:29

7 1.8 Homework 8. Due Friday, Marh 4, Define H sc = (H p +H ac ). (So we have the decomposition H = H p H ac H sc.) Show that if x H sc x 0 then the measure B µ x (B) := E(B)x, x = E (B) x 2 has no atoms yet there exists a Borel set B of Lebesgue measure zero such that E(B)x Show that the decomposition of part c) reduces A. That is, AH i H i, for i = p, ac, or sc. Exercise 1.44 (Behavior of the resolvent near an isolated eigenvalue). We saw in the proof of Theorem?? in Chapter?? that if A is a bounded operator on a complex Banach space λ 0 is not in σ(a) then (A λ) 1 has a power series expansion: (A λ) 1 = n=0 (λ λ 0) n B n valid in some disk λ λ 0 < ε, where each B n is a bounded operator. 1. Suppose that A is the operator on the two dimensional Hilbert space C 2 given by the two by two matrix ( ) 3 1 A =. 0 3 As you (had better) know, σ(a) = {3}. Show that the resolvent (A λ) 1 has a Laurent expansion near λ = 3 with a pole of order two. That is (A λ) 1 = (λ 3) 2 B 2 + (λ 3) 1 B 1 + (λ λ 0 ) n B n which is valid in some punctured disk 0 < λ 3 < a. Find B 2 B 1 show that neither operator is zero. 2. Suppose now that A is a bounded self-adjoint operator on a complex, separable, Hilbert space H. Suppose that λ 0 is an isolated eigenvalue of A, by which we mean that, for some ε > 0 n=0 σ(a) {λ C : λ λ 0 < ε} = {λ 0 }. Prove that (A λ) 1 has a pole of order one around λ 0, in the sense that, for some δ > 0, (A λ) 1 = (λ λ 0 ) 1 B 1 + (λ λ 0 ) n B n, 0 < λ λ 0 < δ, n=0 where the operators B j, j = 1, 0, 1,... are bounded operators on H. Express B 1 in terms of the spectral resolution of A. Exercise Let A be a bounded Hermitian operator on a separable Hilbert space H. Denote by E( ) its spectral resolution. Assume that A 0 write P = E({0}) (which may or may not be the zero projection). Prove that for any vector u in H lim t + e ta u = P u. Exercise Suppose that (X, µ) is a measure space that µ(x) <. Let T : L 2 (µ) L 2 (µ) be a bounded operator. Suppose that range T is contained in L 5 (µ). Show that T is bounded as an operator from L 2 (µ) into L 5 (µ). Hint: Use the closed graph theorem. (See Rudin, Chapter 5, Problem 16. The solution to this problem depends on Theorem 5.10 of Rudin.) Exercise Let T : X X be a densely defined closed operator. If λ, µ ρ (T ), show [T R λ, T R µ ] = 0. Exercise Let H = L 2 (0, 1), D = C([0, 1]). Let (Af)(x) := f (0) 1 where 1 is the constant function, 1. Then A : D H is a densely defined operator. Show D (A ) is not dense. In fact show D (A ) = {1} A f = 0 for all f D (A ). Exercise Suppose that A : H K is a densely defined operator B : H K is a bounded operator. Show (A + B) = A + B. Exercise Let (Ω, F, µ) be a measure space with no infinite atoms ρ, q : Ω C be two measurable functions. Show that M ρ = M q as (unbounded) operators on L 2 (µ) iff ρ = q a.e. Exercise In this exercise we suppose that n N, H = L 2 (R n, Lebesgue), S = S (R n ) is the n dimensional Schwartztest function of smooth rapidly decaying functions. Given N N 0 {a α C : α N} C, let Lf = a α α f for all f D (L) = S Show; α N L f = α N ( 1) α ā α α f for all f D ( L ). 1. Lf, g = f, L g for all f, g S. 2. Show L = L. (Hint: use the Fourier transform to diagonalize L L.) 3. Conclude that L is essentially self-adjoint iff L = L, i.e. a α = ( 1) α ā α for all α N. Page: 7 job: 241ahmsolns macro: svmonob.cls date/time: 5-Feb-2016/11:29

8 [For example, this exercise shows = n essentially self-adjoint.] k=1 2 / x 2 k with D ( ) = S is Exercise Let n = 1 so that S = S (R) further let D := {f S : f = 0 near 0}, where the neighborhood of 0 where f = 0 can vary with f. Then define Af := d2 f for f D (A) := D. dx2 1. Prove that A is densely defined symmetric but is not essentially self 1 adjoint. Hint: Let ϕ = Fourier transform of t 2 i show that A ϕ = iϕ. 2. Find the spectrum of A. Hint: see Conway Theorem, X.2.8 on page 311 to help you guess the answer if it is not clear. Exercise Suppose that A is a linear transformation in H that A is densely defined, closed, one to one, has dense range. Then clearly A 1 exists is densely defined, A (A 1 ) both exist. Prove 1. Nul (A ) = 0, 2. Ran A is dense in H, 3. (A 1 ) = (A ) 1 (which exists by 1. 2.), 4. A 1 is closed.