Exercises 2

Exercise 2.1:

1. Let $T_n \rightarrow T$ uniformly. Prove that $T_n^* \rightarrow T^*$ uniformly.
2. Let $T_n \rightarrow T$ strongly. Give an example where the strong convergence $T_n^* \rightarrow T^*$ does not hold.
3. Let $T_n \rightarrow T$ weakly. Prove that $T_n^* \rightarrow T^*$ weakly.

Exercise 2.2:

Let $P_n$ be a sequence of orthogonal projections and let $P$ be an orthogonal projection. Prove that if $P_n \rightarrow P$ weakly, then $P_n \rightarrow P$ strongly.

Exercise 2.3:

Find a projection on $\mathbb{C}^2$ that is not an orthogonal projection. Find a projection in $C[0,1]$ with the $\sup$ norm. Can you find an orthogonal projection?

Exercise 2.4:

Let $T$ in $L^2(\mathbb{R})$ be the operator of multiplication by a function $t$. Under what condition on $t$ is the operator $T$ an orthogonal projection? Describe $\text{Ran}(T)$ in this case.

Exercise 2.5:

Show that a bounded operator $T$ in $\mathcal{H}$ is normal if and only if the operators $\text{Re}(T) = \frac{(T+T^*)}{2}$ and $\text{Im}(T) = \frac{T-T^*}{2i}$ commute.

Exercise 2.6:

Let $U \in \mathcal{B}(\mathcal{H})$ be such that $\|Ux\| = \|x\|$ for all $x$ and $\text{Ran}(U)=\mathcal{H}$. Prove that $U$ is unitary.

Exercise 2.7:

Let $P,Q$ be orthogonal projections onto subspaces $E,F \subset \mathcal{H}$ respectively. Suppose that $PQ=QP$.

1. Prove that $I-P; PQ; PQ; P+Q - PQ$ and $P+Q-2PQ$ are all orthogonal projections.
2. How are the ranges of all these projections related to $E$ and $F$?.

Exercise 2.8:

Let $A=A^* \in \mathcal{B}(\mathcal{H})$. Prove that if $A \neq 0$, then $A^n \neq 0$ for all $n \in \mathbb{N}$. _Hint: First prove it for $n=2^k, k \in \mathbb{N}$.

Exercise 2.9:

An operator $T \in \mathcal{B}(\mathcal{H})$ is called isometric if $T^*T=I$ (but not necessarily $TT^*=I$). Show that if $\dim \mathcal{H} < \infty$, then all isometric operators are unitary. Show that all isometric operators are injective.