6710-2025-01

Show that for the sphere \(S(0, 1)\) in a real normed space \(X\) and any point \(x_0 \in S(0, 1)\) there is a hyperplane \(H_0 \ni x_0\) such that the closed ball \(\widetilde {B}(0, 1)\) lies entirely in one of the two closed half spaces determined by \(H_0\).

Consider the operator \(T : L^2[0, 1] \to L^2[0, 1]\) defined by

Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuum spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\) and resolvent set \(\rho (T)\).

Let \(T : X \to X\) be a compact linear operator on a normed space \(X\). Show that for every \(\lambda \neq 0\) the null space \(N(T - \lambda I)\) is finite dimensional.

Let \(a\) and \(b\) be non-zero (but possibly complex) elements in the complex Hilbert space \(\ell ^2\). Consider the bounded linear operator \(T : \ell ^2 \to \ell ^2\) defined by

1. Use the Fredholm Alternative Theorem to find a simple condition on \(a\) and \(b\) guar- anteeing that the equation

   \[(T - I)u = v\]

   has a unique solution \(u\) for all \(v \in \ell ^2\).

2. Find the Hilbert adjoint operator \(T^* : \ell ^2 \to \ell ^2\).

3. Given \(T : X \to Y\) a bounded linear operator, recall that the adjoint operator \(T^\times : Y'\to X'\) is defined

   \[(T^\times g)(x) = g(Tx).\]

   Find the adjoint operator \(T^\times : \ell ^2 \to \ell ^2\). Hint: Express \(g\) and \(T^\times g\) using the Riesz Representation Theorem.

4. Use the Fredholm Alternative Theorem to give conditions on \(v_0\) guaranteeing that

   \[(T - I)u = v_0\]

   has a unique solution \(u\). When it exists, find the solution.