Departement of Mathematics University of Utah Real and Complex Analysis Qualifying Exam

Let \(\ell ^1\) be the Banach space of real sequences \(a=(a_1,a_2,\ldots )\) with norm \[ \lVert a\rVert _1=\sum _{i=1}^{\infty }|a_i|<\infty , \] and \(\ell ^\infty \) the Banach space of real sequences with norm \(\lVert x\rVert _\infty =\sup _i |x_i|\). Let \(a\in \ell ^1\) be a fixed sequence; for any sequence \(x\in \ell ^\infty \), define a new sequence \(T_a(x)\) by \[ \bigl (T_a(x)\bigr )_n=\sum _{i=1}^{n}a_i x_i \]

(a) Show that this defines a bounded operator \(T_a:\ell ^\infty \to \ell ^\infty \).

(b) Show \(\lVert T_a\rVert =\lVert a\rVert _1\), where the left-hand side denotes the norm of the operator \(T_a:\ell ^\infty \to \ell ^\infty \).

Let \((\mathcal M,X,\mu )\) be a finite positive measure space.

(a) Show that \(L^2(X)\subset L^1(X)\), and that the inclusion \(i:L^2(X)\subset L^1(X)\) has norm \(\lVert i\rVert =\sqrt {\mu (X)}\).

(b) Let \(\mathcal M'\subset \mathcal M\) be the subset of measurable sets \(E\) with \(\mu (E)>0\). Show that if \[ L^1(X)=L^2(X), \] then \[ \inf _{E\in \mathcal M'}\mu (E)>0. \]

Let \(H\) be a Hilbert space with inner product written as \(\langle f,g\rangle \), and let \(f,g\in H\) be two non-zero elements. Show that \(f=zg\) for some \(z\in \mathbb C^*\) if and only if there exists no \(h\in H\) with \[ \langle f,h\rangle =1 \quad \text {and}\quad \langle g,h\rangle =0. \]

Let \(f:[0,1]\to \mathbb R_{\ge 0}\) be a measurable integrable function \(f\in L^1([0,1])\) (with respect to the Lebesgue measure). Show that \[ \int _{[0,1]} x f(x)\,dx =\int _{[0,1]}\left (\int _{[y,1]} f(x)\,dx\right )\,dy \]

Let \(T:C_0(\mathbb R)\to C_0(\mathbb R)\) be the translation operator defined by \((T.f)(x)=f(x+1)\). Show that there is no non-zero bounded linear functional \(\Phi :C_0(\mathbb R)\to \mathbb C\) invariant under \(T\), i.e. such that \(\Phi \) satisfies \[ \Phi (f)=\Phi (Tf). \]

Compute the integral \[ \int _0^\infty \left (\frac {\sin x}{x}\right )^2\,dx. \]

Let \(a\) be an isolated singularity of the meromorphic function \(f:\Omega \to \mathbb C\). Prove that if \(a\) is an essential singularity, then in any neighborhood of \(a\), the function \(f\) takes values arbitrarily close to any complex number.

Let \(\alpha >1\) be arbitrary. Show that the equation \[ \alpha -z-e^{-z}=0 \] has exactly one solution in the half plane \(\{z:\operatorname {Re}z>0\}\), and moreover, this solution is real.

Let \(\Omega \subsetneq \mathbb C\) be a simply connected region, and fix \(z_0\in \Omega \). If \(\phi :\Omega \to D\) is a conformal map such that \(\phi (z_0)=0\), show that \[ |\phi '(z_0)| =\sup \{|f'(z_0)|:\ f:\Omega \to D\text { holomorphic}, f(z_0)=0\}. \]

Let \(f\) be a function holomorphic on \(D\) and continuous on \(\overline {D}\). Assume that \(|f(z)|=1\) whenever \(|z|=1\). Show that \(f\) can be extended to a meromorphic function on the whole \(\mathbb C\), with at most finitely many poles.

(Hint: Starting with the Schwarz reflection principle for the upper half plane, deduce that an appropriate reflection continuation that can be used in this setting is \(z\mapsto \frac {1}{f(1/\bar z)}\).)