Exam
6710-2022-01
Spring 2022 · 5 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Assume a bounded, self-adjoint operator \(T : H \to H,\) where \(H\) is a Hilbert space, satisfies
\begin{equation} \langle T x, x\rangle \ge \beta \|x\|^2, \tag {1} \end{equation}for some \(\beta > 0\). The goal of this problem is to show that \(T\) is one-to-one and onto.
- Show that the nullspace of \(T\) is trivial, i.e. \(\mathcal {N}(T) = \{0\}\).
- Show that the range \(\mathcal {R}(T)\) is closed.
- Show that \(\mathcal {R}(T)^\perp = \{0\}\).
- Why do we have \(\mathcal {R}(T) = H\)?
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Question 2
Let \(T : H \to H\) be a bounded linear operator on a Hilbert space \(H\). Show that \(T\) is compact if and only if \(T^*T\) is compact.
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Question 3
Let \(H\) be a separable Hilbert space with \((e_n)\) being a total orthonormal sequence of \(H\). Let \(T : H \to H\) be a bounded linear operator satisfying
\[\sum _{n=1}^\infty \|T e_n\| < \infty .\]Show that \(T\) is compact by approximating \(T\) by a sequence \((T_k)\) of finite rank operators that converges to \(T\) in an appropriate sense.
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Question 4
Let \(H\) be a complex Hilbert space and let \(T \in \mathcal {B}(H, H)\). The goal of this problem is to prove that \(\|T\|^2 = \|T^*T\|\).
- Show that \(\|T\|^2 \le \|T^*T\|\).
- Show that \(\|T^*T\| \le \|T\|^2\).
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Question 5
Let \(X\) be a normed vector space. Let \(x_0 \in X\) and consider the closed ball \(\widetilde {B}(0,\|x_0\|)\). Use the Hahn-Banach theorem to show that there exists \(f \in X'\) such that \(f(x_0) = 1\) and for any \(x \in \widetilde {B}(0,\|x_0\|)\) we have \(f(x) \le 1\). Do not use Hahn-Banach for convex sets or the separating hyperplane theorem.