querent

(a) Let $𝑓$ be a measurable real-valued function on a finite measure space $(𝑋,\mathrm{ℬ︀},𝜇)$. Define

for $𝑛\in \mathbb{Z}$. Give and prove a (non-trivial) upper and lower estimate of the ${𝐿}^{𝑝}$ norm of $𝑓$, $1\le 𝑝<\infty$, purely in terms of the quantities ${𝑚}_{𝑛(𝑓)}$.

(b) Suppose that $(𝑋,\mathrm{ℬ︀},𝜇)$ is a $𝜎$-finite measure space, $𝐾$ is a measurable function on $𝑋\times 𝑋$, and

$𝜇$-a.e. Show that the integral operator $𝐴:{𝐿}^2(𝑋)\to {𝐿}^2(𝑋)$ defined by

is well-defined and bounded, and its norm is bounded by $𝐶$.

Write a real number $𝑥\in [0,1)$ in the usual decimal expansion (pick the representation ending in 0's if there are two representations), $𝑥=0.{𝑥}_1{𝑥}_2{𝑥}_3\dots$. We let $𝐴$ be the set of $𝑥\in [0,1)$ with the property that there are infinitely many $𝑛\in \mathbb{N}$ such that each of the digits $0,\dots ,9$ appears among the first $10𝑛$ digits (i.e. ${𝑥}_1,\dots ,{𝑥}_{10𝑛}$) exactly $𝑛$ times. Prove that the set $𝐴$ is Lebesgue measurable and find its measure.

Suppose that $𝑋$ is a complex Banach space and $\mathrm{𝒯︀}$ is its weak topology.

(a) Suppose that $(𝑋,\mathrm{𝒯︀})$ is first countable. Show that there are linear functionals ${𝑓}_{𝑗}\in {𝑋}^{\ast }$, $𝑗=1,2,\dots$, such that every $𝑓\in {𝑋}^{\ast }$ is a finite linear combination of the ${𝑓}_{𝑗}$. That is, if $𝑓\in {𝑋}^{\ast }$ then there exists $𝑁>0$ and ${𝑎}_{𝑗}\in ℂ$, $𝑗=1,\dots ,𝑁$, such that $𝑓={\sum }_{𝑗=1}^{𝑁}{𝑎}_{𝑗}{𝑓}_{𝑗}$.

(b) Suppose that $𝑋$ is infinite dimensional. Show that $(𝑋,\mathrm{𝒯︀})$ is not metrizable.

Suppose that $𝐻$ is a Hilbert space, $𝑇\in \mathrm{ℒ︀}(𝐻)$, and let ${𝑇}^{\ast }$ denote its adjoint.

(a) Show that , where $\oplus$ is orthogonal direct sum.

(b) Suppose that there exists $𝐶>0$ such that for all $𝑥\in 𝐻$, $\Vert 𝑥\Vert \le 𝐶\Vert 𝑇𝑥\Vert$. Show that $Ran(𝑇)$ is a closed subspace of $𝐻$.

(c) Show that if $𝑇{𝑇}^{\ast }=𝐼={𝑇}^{\ast }𝑇$, then $𝑇-𝜆𝐼\in \mathrm{ℒ︀}(𝐻)$ is invertible if $|𝜆|\ne 1$, and show that $\Vert {(𝑇-𝜆𝐼)}^{-1}\Vert \le {|1-|𝜆||}^{-1}$.

We define a bounded operator $𝐴:{\ell }^2(\mathbb{Z})\to {\ell }^2(\mathbb{Z})$ by

(a) Show that $𝐴$ is a bounded symmetric operator.

(b) Let $𝑇:{\ell }^2(\mathbb{Z})\to {𝐿}^2([-𝜋,𝜋])$ be defined by

Show that the operator $𝑇𝐴{𝑇}^{-1}:{𝐿}^2([-𝜋,𝜋])\to {𝐿}^2([-𝜋,𝜋])$ is a multiplication operator; that is,

for some function $𝜇(𝑡)$.

(c) Determine the spectrum of $𝐴$.

(d) Find the eigenvalues of $𝐴$.

Let ${\mathrm{\Omega }}_{+}=\{𝑧\in ℂ:0<Im𝑧<1\}$, ${\mathrm{\Omega }}_{-}=\{𝑧\in ℂ:-1<Im𝑧<0\}$. Let $\mathrm{𝒮︀}(ℝ)$ denote the space of Schwartz functions on $ℝ$, with seminorms , and ${\mathrm{𝒮︀}}^{\prime }(ℝ)$ its topological dual, tempered distributions.

(a) Suppose that ${𝑢}_{+}:{\mathrm{\Omega }}_{+}\to ℂ$ is an analytic function with $|{𝑢}_{+}(𝑧)|\le 𝐶({|Im𝑧|}^{-𝑘}+{|Re𝑧|}^{\ell }+1)$ for some $𝐶,𝑘,\ell$. For $𝜀\in (0,1)$, let ${𝑢}_{+,𝜀}\in {\mathrm{𝒮︀}}^{\prime }(ℝ)$ with ${𝑢}_{+,𝜀}(𝜑)={\int }_{ℝ}𝑢(𝑥+𝑖𝜀)𝜑(𝑥)d𝑥$. Show that ${𝑢}_{+,0}={lim}_{𝜀\to 0^{+}}{𝑢}_{𝜀}$ exists in ${\mathrm{𝒮︀}}^{\prime }(ℝ)$. (Hint: consider the indefinite integral of $𝑢$ from e.g. ${𝑧}_0=𝑖/2$, and integrate first parallel to the real axis then to the imaginary axis and obtain an estimate for ${\int }_{{𝑧}_0}^{𝑧}𝑢(𝑤)d𝑤$.) Define ${𝑢}_{-,0}$ similarly, replacing ${\mathrm{\Omega }}_{+}$ by ${\mathrm{\Omega }}_{-}$.

(b) For ${𝑢}_{\pm }(𝑧)={𝑧}^{-𝑚}$, $𝑧\in {\mathrm{\Omega }}_{\pm }$, $𝑚\ge 1$ integer, find ${𝑢}_{+,0}(𝜑)-{𝑢}_{-,0}(𝜑)$, $𝜑\in \mathrm{𝒮︀}(ℝ)$, in terms of , $𝑗\ge 0$.

Two short problems.

(a) Suppose that $(𝑋,\mathrm{𝒜︀},𝜇)$ is a finite measure space and $𝑓:𝑋\to ℝ$ is an $\mathrm{𝒜︀}$-measurable function. Prove that ${\Vert 𝑓\Vert }_{{𝐿}^{\infty }}={lim}_{𝑝\to \infty }{\Vert 𝑓\Vert }_{{𝐿}^{𝑝}}$.

(b) Let $\top =ℝ/(2𝜋\mathbb{Z})$, and identify it with the unit circle $\{𝑧\in ℂ:|𝑧|=1\}$ in the complex plane. For $𝛿\in (0,1)$, let ${\mathrm{\Omega }}_{𝛿}=\{𝑧\in ℂ:1-𝛿<|𝑧|<1\}$. Suppose that $𝜑\in 𝐶(\top )$ and there exist $𝛿\in (0,1)$ and , $𝑓$ holomorphic on ${\mathrm{\Omega }}_{𝛿}$, such that $𝑓{|}_{\top }=𝜑$. Let

be the $𝑛$th Fourier coefficient of $𝜑$. Show that for all $𝑘>0$ there exists $𝐶>0$ such that $|{(\mathrm{ℱ︀}𝜑)}_{𝑛}|\le 𝐶{|𝑛|}^{-𝑘}$ for $𝑛\le -1$.

Suppose $𝐴\subseteq ℝ$ is Borel, $𝑇$ is a dense subset of $ℝ$ and ${𝜏}_{𝑡}(𝐴)-𝐴$ has Lebesgue measure zero for each $𝑡\in 𝑇$, where ${𝜏}_{𝑡}:ℝ\to ℝ$ is the translation $𝑥\to 𝑥+𝑡$. Prove that either $𝐴$ or $ℝ-𝐴$ has Lebesgue measure zero.

Suppose that $𝑓:\mathbb{N}\to ℂ$ is a bounded function (where $\mathbb{N}$ is the set of the positive integers), and define ${𝑀}_{𝑓}\in \mathrm{ℒ︀}({\ell }^{𝑝})$, $1\le 𝑝<\infty$, by ${𝑀}_{𝑓}({\{{𝑎}_{𝑛}\}}_{𝑛=1}^{\infty })={\{𝑓(𝑛){𝑎}_{𝑛}\}}_{𝑛=1}^{\infty }$.

(a) Find (with proof) the spectrum of ${𝑀}_{𝑓}$.

(b) State and prove a necessary and sufficient condition in terms of $𝑓$ for ${𝑀}_{𝑓}$ being a compact operator.

(c) Can there be any points in the spectrum of ${𝑀}_{𝑓}$ that are not eigenvalues? Give an example or prove the contrary.

Two short problems.

(a) Let $\top =ℝ/(2𝜋\mathbb{Z})$ be the circle, and let ${𝐶}^1(\top )$ denote the Banach space of complex valued continuously differentiable functions on $\top$ with the norm ${\Vert 𝑓\Vert }_{{𝐶}^1}=sup|𝑓|+sup|{𝑓}^{\prime }|$. Suppose that $𝑢:{𝐶}^1(\top )\to ℂ$ is continuous and has the property that $𝜑\in {𝐶}^1(\top )$, $𝜑\ge 0$ on $\top$, imply $𝑢(𝜑)\ge 0$. Show that there is a finite Borel measure $𝜇$ on $\top$ such that $𝑢(𝜑)=\int 𝜑d𝜇$ for all $𝜑\in {𝐶}^1(\top )$.

(b) Suppose that $𝐻$ is a Hilbert space, $\mathrm{ℒ︀}(𝐻)$ the set of bounded operators on $𝐻$, $𝐴\in \mathrm{ℒ︀}(𝐻)$ is self-adjoint, and for $𝑡\in ℝ$ define $𝑈(𝑡)={𝑓}_{𝑡}(𝐴)$ via the functional calculus where ${𝑓}_{𝑡}(𝑠)={𝑒}^{𝑖𝑡𝑠}$. Show that $𝑈(𝑡)-𝐼$ is compact for all $𝑡\in ℝ$ if and only if $𝐴$ is compact.