An Introduction to the Invariant Subspace Problem

Verify that the Hermitian matrices (as defined in Exercise 5.2.5) are exactly the Hermitian operators on the finite-dimensional Hilbert space $\mathbb {C}^{n}$.

Describe the Fredholm integral operators of the first kind that are Hermitian.

Let $K$ be a subspace of a Hilbert space $H$.

Complete the proof of Theorem 5.15 by verifying the uniqueness assertion.

Theorem 5.15: Assume that $K$ is a closed subspace of a Hilbert space $H$ and that $x \in H$. Then there exists $y \in K$ such that $\left\| x - y\right\| = \inf \left\{ \left\| x - z\right\| \mid z \in K\right\}$. Further, the element $y$ is the unique element of $K$ with this property.

Prove Theorem 5.17.

Theorem 5.17: Let $H$ be a Hilbert space. If $T \in \mathcal{B}(H)$ is Hermitian, then $\ker (T)$ and $\operatorname {range}(T)$ are subspaces of $H$ and satisfy $\ker (T)^{\perp } = \operatorname {range}(T)$ and $\operatorname {range}(T)^{\perp } = \ker (T)$ (here $\ker (T)$ and $\operatorname {range}(T)$ denote, respectively, the kernel and range of the map $T$).

It follows from Theorem 5.18 that any eigenvalue of a Hermitian operator is real-valued. Give a more direct proof of this fact.

Theorem 5.18: Let $H$ be a Hilbert space. If $T \in \mathcal{B}(H)$ is Hermitian, then $\sigma (T) \subseteq \mathbb {R}$.

Let $T$ be the Volterra operator on $L^{2}([0,1])$ defined by

For $0 \leq \alpha \leq 1$ set