PhD Preliminary Qualifying Examination Applied Mathematics January 7 2015

Let \(M\) be a subset of a Hilbert space \(H\). The goal of this problem is to show that: \[ \overline {\operatorname {span}M}=H \quad \text {if and only if}\quad M^\perp =\{0\}. \]

1. Assume \(\overline {\operatorname {span}M}=H\) and let \(x\in M^\perp \). Show that \(x=0\).
2. Assume \(M^\perp =\{0\}\). Show that \(\overline {\operatorname {span}M}=H\).

Let \((x_n)\) be a sequence in a Hilbert space \(H\). Show that: \[ x_n\to x\ \textbf {strongly} \quad \text {if and only if}\quad (x_n\to x\ \textbf {weakly}\text { and }\lVert x_n\rVert \to \lVert x\rVert ). \]

Let \(u,v\) be non-zero elements of a Hilbert space \(H\). Consider the linear operator \(T:H\to H\) defined for \(x\in H\) by \(Tx=u(x,v)\).

1. Briefly explain why the operator \(T\) is compact.
2. Find a condition on \(u\) and \(v\) that guarantees that the equation \[ (I-T)x=y \] admits a unique solution \(x\) for all \(y\in H\).

Let \((\lambda _n)\in \ell ^2\) and consider the operator \(T:\ell ^2\to \ell ^2\) defined by \(y=Tx\), where \(x=(\xi _j)\in \ell ^2\), \(y=(\eta _j)\in \ell ^2\) and \[ \eta _j=\sum _{k=1}^{\infty }\alpha _{jk}\xi _k,\qquad j=1,2,\ldots , \] where the \(\alpha _{jk}\) satisfy \[ \sum _{j=1}^{\infty }\sum _{k=1}^{\infty }\lvert \alpha _{jk}\rvert ^2<\infty . \] Show that \(T\) is a compact operator. Hint: Approximate \(T\) with finite rank operators \(T_n\).

Let \(T:X\to X\) be a bounded linear operator defined on a Banach space \(X\). Denote by \(R_\lambda \) the resolvent operator associated with \(T\) for some \(\lambda \in \rho (T)\). The resolvent set of \(T\) is denoted by \(\rho (T)\).

1. Use Neumann series to show that for \(\lambda ,\mu \in \rho (T)\), \[ R_\lambda =\sum _{j=0}^{\infty }(\lambda -\mu )^j R_\mu ^{j+1}, \] where the series is absolutely convergent in the operator norm when \[ \lvert \lambda -\mu \rvert <\lVert R_\mu \rVert ^{-1}. \]
2. Deduce from part (a) whether the spectrum \(\sigma (T)\) is open, closed or neither.

Find the radius of convergence of the Taylor series with the center at the origin (\(x_0=0\)) for the following function

1. \[ f(x)=\frac {\sqrt {x+3}}{x^2+3} \]
2. \[ f(x)=\frac {1}{(\cos x+3)^2} \]

(\(x\) is a real variable; you do not need to find the series themselves).

Let \(D_1\) and \(D_2\) be two disjoint domains, whose boundaries share a common curve \(\Gamma \). Let

• \(f(z)\) be analytic in \(D_1\) and continuous in \(D_1\cup \Gamma \)
• \(g(z)\) be analytic in \(D_2\) and continuous in \(D_2\cup \Gamma \)
• \(f(z)=g(z)\) for \(z\in \Gamma \)

Show that the function \[ H(z)= \begin {cases} f(z), & z\in D_1,\\ f(z)=g(z), & z\in \Gamma ,\\ g(z), & z\in D_2 \end {cases} \] is analytic in \(D=D_1\cup \Gamma \cup D_2\).

[Hint: Use the Morera theorem.]

1. Show that any analytic function (not identically equal to zero) can have only isolated zeros inside its analyticity domain.
2. Can an analytic function have a non-isolated singularuty?
3. Prove the Uniqueness Theorem: If two functions are analytic in a domain \(D\) and equal on some set of points that has a limiting point inside \(D\), then these functions are identically equal in \(D\).

Evaluate

1. \[ \int _{-\infty }^{\infty } e^{ix^2}\,dx \]
2. \[ \int _0^\infty \frac {\sin x}{x}\,dx. \]

Find the leading behavior, as \(s\to +\infty \), of the integral

1. \[ I(s)=\int _0^3 \frac {1}{\sqrt {x^2+2x}}e^{-sx}\,dx \]
2. \[ I(s)=\int _0^{\pi /2} e^{is\cos x}\,dx \]