UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph.D. Preliminary Examination in Applied Mathematics August 16th, 2017.

Let \(H\) be a Hilbert space. Show that an isometric linear operator \(T:H\to H\) satisfies \(T^*T=I\), where \(I\) is the identity operator on \(H\).

Let \((X,d)\) be a complete metric space and \(f\) a contraction on \(X\) with contraction constant \(\alpha <1\). Let \(x_f\) be the unique fixed point of \(f\). Fix \(x_0\in X\) and consider the sequence of points defined by \(x_n=f(x_{n-1})\) for \(n\in \mathbb {N}\).

1. Prove that \[ d(x_n,x_f)\le \frac {\alpha ^n}{1-\alpha }d(x_0,x_1). \] Hint: consider \(d(x_m,x_n)\) for \(m>n\).
2. Let \(S=[1,\infty )\) and \(f:S\to S\) be defined by \(f(x)=\frac {1}{2}(x+2/x)\). Prove that \(f\) is a contraction mapping on \(S\) and compute a contraction constant. Determine all fixed points of \(f\) on \(S\). Using part (a), explain why the sequence \((x_n)\) might be useful.

Let \(X\) be an inner product space with inner product \(\langle \cdot ,\cdot \rangle \). Let \((a_j)\) be a sequence of complex numbers and let \((\psi _j)\) and \((\phi _j)\) be two orthonormal sequences in \(X\). Let \(T:X\to X\) be an operator defined by \[ Tx=\sum _{j=1}^{\infty } a_j\langle \psi _j,x\rangle \phi _j. \] State and prove a (nontrivial) sufficient condition on \((a_j)\) so that \(T\) is a compact operator.

The right-shift operator \(R:\ell ^2\to \ell ^2\) is defined by \(R(\xi _1,\xi _2,\ldots )=(0,\xi _1,\ldots )\) and the left-shift operator \(L:\ell ^2\to \ell ^2\) is defined by \(L(\xi _1,\xi _2,\ldots )=(\xi _2,\xi _3,\ldots )\).

1. Show that \(L\) and \(R\) are adjoint to each other.
2. Show that \(\lVert R\rVert =\lVert L\rVert =1\).
3. Prove that the spectrum of \(L\) is given by \[ \sigma _p(L)=\{\lambda \in \mathbb {C}:\lvert \lambda \rvert <1\},\quad \sigma _c(L)=\{\lambda \in \mathbb {C}:\lvert \lambda \rvert =1\},\quad \sigma _r(L)=\emptyset . \]
4. Prove that the spectrum of \(R\) is given by \[ \sigma _p(R)=\emptyset ,\quad \sigma _c(R)=\{\lambda \in \mathbb {C}:\lvert \lambda \rvert =1\},\quad \sigma _r(R)=\{\lambda \in \mathbb {C}:\lvert \lambda \rvert <1\}. \]

Let \(X\) be a complex Banach space and \(T\in B(X,X)\). Show that the resolvent, \(R_\lambda (T)\), satisfies \(\lVert R_\lambda (T)\rVert \to 0\) as \(\lambda \to \infty \).

1. State and prove Liouville’s theorem.
2. Show that every polynomial of degree \(n\ge 1\) in the complex plane has at least one root.

Find the real-valued function \(\Phi (z)\) that solves Laplace’s equation in the region between the two noncentric circles \(\lvert z\rvert =1\) and \(\lvert z-1\rvert =\frac {5}{2}\) subject to the boundary conditions \(\Phi =0\) on \(\lvert z\rvert =1\) and \(\Phi =1\) on \(\lvert z-1\rvert =\frac {5}{2}\).

Consider the transformation on the extended complex plane \(\overline {\mathbb {C}}\) \[ f(z)=w=z+\frac {1}{z}. \]

1. Find all possible fixed points.
2. Show that the image of the points in the upper half \(z\)-plane that are exterior to the circle \(\lvert z\rvert =1\) corresponds to the entire upper half \(w\)-plane.
3. Find the image of the unit circle.
4. Show that \(f(z)\) is a two-to-one function.

Evaluate the integral \[ \int _0^\infty \frac {\sin (ax)}{x}\,dx \qquad \text {for }a>0. \] You must clearly show the contours of integration and treat integration over each contour separately.

Consider the Dawson’s integral \[ D(x)=e^{-x^2}\int _0^x e^{t^2}\,dt \qquad \text {as }x\to \infty . \]

1. Find the first non-constant term of the asymptotic expansion of \(D(x)\) and show that the relationship is asymptotic.
2. Find the full asymptotic series of \(D(x)\).

HINT: Introduce a cutoff parameter.