querent

Spam yourself with math PhD qualifying-exam questions.

Stanford Fall 2011, 3 Real analysis, bounded operators, Hilbert space geometry

In this problem $𝑋, 𝑌$ are Hilbert spaces, $ℒ︀ (𝑋, 𝑌)$ the set of bounded linear operators from $𝑋$ to $𝑌$ . Prove or disprove the following statements.

(a) There exists $𝑇 \in ℒ︀ (𝑋, 𝑌)$ such that $𝑇$ is a bijection, but the set theoretic inverse $𝑇^{- 1}$ is not in $ℒ︀ (𝑌, 𝑋)$ .

(b) There exists $𝑇 \in ℒ︀ (𝑋, 𝑌)$ such that $𝑇$ is injective, but there is no left inverse $𝑆 \in ℒ︀ (𝑌, 𝑋)$ for $𝑇$ (i.e. there is no $𝑆 \in ℒ︀ (𝑌, 𝑋)$ such that $𝑆 𝑇$ is the identity on $𝑋$ ).

(c) There exists $𝑇 \in ℒ︀ (𝑋, 𝑌)$ such that $𝑇$ is surjective, but there is no right inverse $𝑆 \in ℒ︀ (𝑌, 𝑋)$ for $𝑇$ (i.e. there is no $𝑆 \in ℒ︀ (𝑌, 𝑋)$ such that $𝑇 𝑆$ is the identity on $𝑌$ ).

Stanford Fall 2011, 3 Real analysis, bounded operators, Fourier transform, spectral theory

Let $ℒ︀ (𝐿^{2} (ℝ))$ denote the set of bounded linear operators on $𝐿^{2} (ℝ)$ . Consider the following operators $Λ_{𝑠}$ , $𝑠 \in ℝ$ , on $𝐿^{2} (ℝ)$ : $(Λ_{𝑠} 𝑓) (𝜉) = {(1 + {| 𝜉 |}^{2})}^{𝑖 𝑠 / 2} 𝑓 (𝜉)$ . Prove or disprove each of the following statements for the map $Λ : ℝ \to ℒ︀ (𝐿^{2} (ℝ))$ given by $Λ (𝑠) = Λ_{𝑠}$ :

(a) $Λ$ is continuous when $ℒ︀ (𝐿^{2} (ℝ))$ is equipped with the norm topology.

(b) $Λ$ is continuous when $ℒ︀ (𝐿^{2} (ℝ))$ is equipped with the strong operator topology.

Stanford Fall 2011, 4 Real analysis, reflexivity, weak topology, weak sequential compactness

Suppose that $𝑋$ is a separable reflexive Banach space.

(a) Show that $𝐵 = {𝑥 \in 𝑋 : ‖ 𝑥 ‖ \leq 1}$ is compact in the weak topology on $𝑋$ .

(b) Give (and prove) a necessary and sufficient condition for ${𝑥 \in 𝑋 : ‖ 𝑥 ‖ = 1}$ to be compact in the weak topology on $𝑋$ .

(c) Prove that $𝐵 = {𝑥 \in 𝑋 : ‖ 𝑥 ‖ \leq 1}$ is sequentially compact in the weak topology of $𝑋$ , i.e. every sequence in $𝐵$ has a weakly convergent subsequence.

Stanford Fall 2011, 4 Real analysis, distributions, weak-star topology

Let ${𝒟︀}^{'} (⊤)$ denote the set of distributions on $⊤ = ℝ / (2 𝜋 ℤ)$ , i.e. the dual of $𝐶^{\infty (⊤)}$ , equipped with the "weak-* topology", i.e. the weakest topology in which the maps $𝐸_{𝜑} : {𝒟︀}^{'} (⊤) \to ℂ$ , $𝐸_{𝜑} (𝑢) = 𝑢 (𝜑)$ are continuous for all $𝜑 \in 𝐶^{\infty (⊤)}$ .

(a) Show that the multiplication map $𝑀 : 𝐶 (⊤) \times 𝐶 (⊤) \to 𝐶 (⊤)$ given by $(𝑀 (𝑓, 𝑔)) (𝑥) = 𝑓 (𝑥) 𝑔 (𝑥)$ has no continuous extension to a map $\tilde{𝑀} : {𝒟︀}^{'} (⊤) \times {𝒟︀}^{'} (⊤) \to {𝒟︀}^{'} (⊤)$ .

(b) Show that, on the other hand, the convolution map $* : 𝐶 (⊤) \times 𝐶 (⊤) \to 𝐶 (⊤)$ given by $(𝑓 * 𝑔) (𝑥) = \frac{1}{2 𝜋} \int_{0}^{2 𝜋} 𝑓 (𝑥 - 𝑦) 𝑔 (𝑦) d 𝑦$ has a unique (separately) continuous extension to a map $\tilde{*} : {𝒟︀}^{'} (⊤) \times {𝒟︀}^{'} (⊤) \to {𝒟︀}^{'} (⊤)$ .

Stanford Fall 2011, 5 Real analysis, Borel measures, Lebesgue measure, invariant measures

Recall that $SL (2, ℝ) \subseteq GL (2, ℝ)$ , consisting of matrices of determinant 1, acts on $ℝ^{2}$ by matrix-vector multiplication $(𝐴, 𝑥) \to 𝐴 𝑥 \in ℝ^{2}$ . Find all Baire measures $𝜇$ on $ℝ^{2}$ (i.e. Borel measures finite on compact sets) which are invariant under the $SL (2, ℝ)$ -action, i.e. such that $𝜇 (𝑆) = 𝜇 (𝐴 𝑆)$ (with $𝐴 𝑆 = {𝐴 𝑥 : 𝑥 \in 𝑆}$ ) for all $𝐴 \in SL (2, ℝ)$ and $𝑆$ Borel.

Stanford Fall 2011, 5 Real analysis, Sobolev spaces, spectral theory, elliptic operators

For $𝑠 \geq 0$ , let $𝐻^{𝑠 (⊤)}$ be the space of $𝐿^{2}$ functions $𝑓$ on the circle $⊤ = ℝ / (2 𝜋 ℤ)$ whose Fourier coefficients ${\hat{𝑓}}_{𝑛} = \int 𝑒^{- 𝑖 𝑛 𝑥} 𝑓 (𝑥) d 𝑥$ satisfy $\sum {(1 + 𝑛^{2})}^{𝑠} {| {\hat{𝑓}}_{𝑛} |}^{2} < \infty$ , and define the norm on $𝐻^{𝑠 (⊤)}$ by ${‖ 𝑓 ‖}_{𝑠}^{2} = {(2 𝜋)}^{- 1} \sum {(1 + 𝑛^{2})}^{𝑠} {| {\hat{𝑓}}_{𝑛} |}^{2}$ .

Consider $𝐴 = - \frac{d^{2}}{d 𝑥^{2}}$ , $𝐴 \in ℒ︀ (𝐻^{2} (⊤), 𝐿^{2} (⊤))$ , suppose $𝑉 \in ℒ︀ (𝐿^{2} (⊤))$ , and let $𝐿 = 𝐴 + 𝑉 \in ℒ︀ (𝐻^{2} (⊤), 𝐿^{2} (⊤))$ . With $𝑅 (𝜆) = {(𝜆 𝐼 - 𝐿)}^{- 1} : 𝐿^{2} (⊤) \to 𝐻^{2} (⊤)$ when $𝜆 𝐼 - 𝐿 : 𝐻^{2} (⊤) \to 𝐿^{2} (⊤)$ is invertible, show that $ℂ ∋ 𝜆 \mapsto 𝑅 (𝜆)$ is a meromorphic operator-valued function, and that there exists a half-plane $Ω = {𝜆 \in ℂ : Re 𝜆 < - 𝐶}$ such that $𝑅 (𝜆)$ is holomorphic on $Ω$ .

Stanford Fall 2012, 1 Real analysis, distributions, uniform boundedness, Lp spaces

Two short problems.

(a) Suppose $𝑢$ is a distribution on $ℝ$ and $𝑥^{𝑘} 𝑢 = 0$ for some $𝑘 \in ℕ$ , $𝑘 \geq 1$ . Show that there exist $𝑎_{𝑗} \in ℂ$ such that $𝑢 (𝜑) = \sum_{𝑗 = 0}^{𝑘 - 1} 𝑎_{𝑗} 𝜑^{(𝑗)} (0)$ .

(b) Suppose that $(𝑋, 𝜇)$ is a measure space, $1 < 𝑝 < \infty$ , $𝑢_{𝑛} \in 𝐿^{𝑝 (𝑋, d 𝜇)}$ for $𝑛 \in ℕ$ , and for all $𝜑 \in 𝐿^{𝑞 (𝑋, d 𝜇)}$ , $𝑞^{- 1} + 𝑝^{- 1} = 1$ , ${lim}_{𝑛 \to \infty} \int_{𝑋} 𝑢_{𝑛} 𝜑 d 𝜇$ exists. Show that there exists $𝐶 \geq 0$ such that $\int_{𝑋} {| 𝑢_{𝑛} |}^{𝑝} d 𝜇 \leq 𝐶$ for all $𝑛$ .

Stanford Fall 2012, 1 Real analysis, spectral theory, Lp spaces

Two short problems.

(a) Show that the spectrum of a bounded linear operator $𝐴$ on a Banach space $𝑋$ is non-empty.

(b) Show that if $1 < 𝑝 < \infty$ then for $𝑓 \in 𝐿^{𝑝 (ℝ^{𝑛})}$ , $𝑔 \in 𝐿^{𝑞 (ℝ^{𝑛})}$ , $𝑝^{- 1} + 𝑞^{- 1} = 1$ , $𝑓 * 𝑔$ defined by $(𝑓 * 𝑔) (𝑥) = \int_{ℝ^{𝑛}} 𝑓 (𝑥 - 𝑦) 𝑔 (𝑦) d 𝑦$ is a bounded continuous function with sup norm $\leq {‖ 𝑓 ‖}_{𝐿^{𝑝}} {‖ 𝑔 ‖}_{𝐿^{𝑞}}$ .

Stanford Fall 2012, 2 Real analysis, Fourier series, trigonometric polynomials

Let $𝐷_{𝑁}$ , $𝑁 \geq 1$ integer, be the Dirichlet kernel

𝐷_{𝑁} (𝜃) = \frac{1}{2 𝜋} \frac{sin (𝑁 + \frac{1}{2}) 𝜃}{sin \frac{1}{2} 𝜃} .

Let $𝐿_{𝑁} = \int_{0}^{2 𝜋} | 𝐷_{𝑁} (𝜃) | d 𝜃$ . Prove that there exist $𝐶_{1}, 𝐶_{2} > 0$ such that for all $𝑁 \geq 2$ ,

𝐶_{1} log 𝑁 \leq 𝐿_{𝑁} \leq 𝐶_{2} log 𝑁 .

Stanford Fall 2012, 2 Real analysis, density, compact operators, Lp spaces

Two short problems.

(a) Let $𝑃$ denote the space of continuous piecewise affine functions on $[0, 1]$ . Show that any $𝑓 \in 𝐶 ([0, 1])$ is a uniform limit of elements of $𝑃$ .

(b) Show that the inclusion map $𝜄 : 𝐶 ([0, 1]) \to 𝐿^{2} ([0, 1])$ is not compact.

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