6710-2024-08

Let \(X\) be a Banach space and \(A\), \(T\) be bounded linear operators from \(X\) to itself. Assume further that \(A\) is compact. Show that \(TA\) and \(AT\) are also compact operators.

Let \(C\) be a closed convex set in a Hilbert space \(H\). Show that \(C\) contains a unique element of minimal norm.

For \(k \in \mathbb {N}\) let \(T_k : \ell ^1 \to \ell ^\infty \) be the linear operator given by

where \(e_j = (0, 0, \ldots , 1, \ldots , 0, 0, \ldots )\), i.e. it is the element of \(\ell ^\infty \) with a single 1 in the \(j\)th position and all other entries are zero.

1. Compute \(\|T_k\|\) for each \(k\), and conclude that each \(T_k\) is bounded.
2. Prove that the sequence \((T_n)\) is strongly operator convergent.
3. Prove that the sequence \((T_n)\) is not uniformly operator convergent.

Let \(H\) be a Hilbert space and \(A : H \to H\) be a bounded self-adjoint linear operator.

1. Prove that

   \[\|A\| = \sup _{x \neq 0} \frac {|\langle Ax, x\rangle |}{\|x\|^2}.\]
2. Prove that at least one of \(\|A\|\) or \(-\|A\|\) is an element of \(\sigma (A)\).

Let \(X\) be a normed space over \(\mathbb {R}\) and \(Y \subset X\) be a linear subspace. For \(x_0 \in X \setminus Y\) we define the linear functional

by

1. Prove that \(f\) is continuous if and only if \(x_0 \notin \overline {Y}\).

2. Prove that \(\overline {Y} = \bigcap \{\operatorname {Ker} g \mid g \in X', Y \subset \operatorname {Ker} g\}\), where \(X'\) is the dual of \(X\).

   Hint for part \((ii)\): The \(\subset \) direction is relatively easy. For the other direction, note that as given in part (i), \(f\) satisfies \(f|_Y = 0\) and \(f(x_0) = 1\). The Hahn-Banach theorem provides an extension of \(f\) to the entire space, use this to prove the other direction by contradiction.