Exam
6710-2024-01
Spring 2024 · 5 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Let \((\lambda _j)\) be a sequence that is dense in \([0, 1]\) and consider the linear operator \(T : \ell ^2 \to \ell ^2\) defined by its action on a sequence \((\xi _j) \in \ell ^2\) by
\[T(\xi _j) = (\lambda _j \xi _j).\]Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuum spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\) and resolvent \(\rho (T)\).
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Question 2
Let \(H\) be a Hilbert space and \(T : H \to H\) be a compact operator. Show that the Hilbert adjoint \(T^*\) is also compact.
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Question 3
Let \(X\) be a normed vector space. Use the Hahn-Banach theorem to show that for any \(x \in X\) we have that
\[\|x\| = \sup _{f \in X', f \neq 0} \frac {|f(x)|}{\|f\|}.\] -
Question 4
Let \(M\) be a subset of a Hilbert space \(H\). Assume \(M\) is such that for any \(v, w \in H\) for which the equality
\[\langle v, x\rangle = \langle w, x\rangle \]holds for all \(x \in M\), we must have \(v = w\). Show that \(M^\perp = \{0\}\).
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Question 5
Let \(X\) be a Banach space, \(f : X \to \mathbb {R}\) be a bounded linear functional, \(y \in X\) be fixed, and \(\alpha \in \mathbb {R}\).
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Prove that there exists a constant \(C > 0\) such that if \(|\alpha | < C\), then the non-linear equation
\begin{equation} x + \alpha f(x)x = y \tag {1} \end{equation}has a unique solution \(x\) in the ball \(B = \{x \in X : \|x - y\| \le 1\}\).
- Suggest an interative procedure for approximating the unique solution \(x\) to equation (1).
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