Projections and Idempotents; Invariant and Reducing Subspaces

Let $\mathcal{H}$ be the two-dimensional real Hilbert space $\mathbb {R}^{2}$, let $\mathcal{M} \equiv \left\{ (x,0) \in \mathbb {R}^{2}: x \in \mathbb {R}\right\}$ and let $\mathcal{N} \equiv \left\{ (x, x \tan \theta ): x \in \mathbb {R}\right\}$, where $0 < \theta < \frac{1}{2}\pi$. Find a formula for the idempotent $E_{\theta }$ with $\operatorname {ran} E_{\theta } = \mathcal{M}$ and $\ker E_{\theta } = \mathcal{N}$. Show that $\left\| E_{\theta }\right\| = (\sin \theta )^{-1}$.

Prove Proposition 3.2 (c): If $\mathcal{M} = \operatorname {ran} E$ and $\mathcal{N} = \ker E$ (where $E$ is an idempotent), then $\mathcal{M} \cap \mathcal{N} = (0)$ and $\mathcal{M} + \mathcal{N} = \mathcal{H}$.

Let $\left\{ \mathcal{M}_{i}: i \in I\right\}$ be a collection of closed subspaces of $\mathcal{H}$ and show that $\cap \left\{ \mathcal{M}_{i}^{\perp }: i \in I\right\} = [\vee \left\{ \mathcal{M}_{i}: i \in I\right\} ]^{\perp }$ and $[\cap \left\{ \mathcal{M}_{i}: i \in I\right\} ]^{\perp } = \vee \left\{ \mathcal{M}_{i}^{\perp }: i \in I\right\}$.

Let $P$ and $Q$ be projections. Show:

Generalize Exercise 4 as follows. Suppose $\left\{ \mathcal{M}_{i}: i \in I\right\}$ is a collection of subspaces of $\mathcal{H}$ such that $\mathcal{M}_{i} \perp \mathcal{M}_{j}$ if $i \neq j$. Let $P_{i}$ be the projection of $\mathcal{H}$ onto $\mathcal{M}_{i}$ and show that for all $h$ in $\mathcal{H}$, $\sum \left\{ P_{i} h: i \in I\right\}$ converges to $Ph$, where $P$ is the projection of $\mathcal{H}$ onto $\vee \left\{ \mathcal{M}_{i}: i \in I\right\}$.

Exercise 4 (context): Let $P$ and $Q$ be projections. (a) $P+Q$ is a projection if and only if $\operatorname {ran} P \perp \operatorname {ran} Q$. If $P+Q$ is a projection, then $\operatorname {ran}(P+Q) = \operatorname {ran} P + \operatorname {ran} Q$ and $\ker (P+Q) = \ker P \cap \ker Q$. (b) $PQ$ is a projection if and only if $PQ=QP$. If $PQ$ is a projection, then $\operatorname {ran} PQ = \operatorname {ran} P \cap \operatorname {ran} Q$ and $\ker PQ = \ker P + \ker Q$.

If $P$ and $Q$ are projections, then the following statements are equivalent.

1. $P-Q$ is a projection.

2. $\operatorname {ran} Q \subseteq \operatorname {ran} P$.

3. $PQ=Q$.

4. $QP=Q$. If $P-Q$ is a projection, then $\operatorname {ran}(P-Q) = (\operatorname {ran} P) \ominus (\operatorname {ran} Q)$ and $\ker (P-Q) = \operatorname {ran} Q + \ker P$.

Let $P$ and $Q$ be projections. Show that $PQ=QP$ if and only if $P+Q-PQ$ is a projection. If this is the case, then $\operatorname {ran}(P+Q-PQ) = \operatorname {ran} P + \operatorname {ran} Q$ and $\ker (P+Q-PQ) = \ker P \cap \ker Q$.

Give an example of two noncommuting projections.

Let $A \in \mathcal{B}(\mathcal{H})$ and let $\mathcal{N} = \operatorname {graph} A \subseteq \mathcal{H} \oplus \mathcal{H}$. That is, $\mathcal{N} = \left\{ h \oplus Ah: h \in \mathcal{H}\right\}$. Because $A$ is continuous and linear, $\mathcal{N} \leq \mathcal{H} \oplus \mathcal{H}$. Let $\mathcal{M} = \mathcal{H} \oplus (0) \leq \mathcal{H} \oplus \mathcal{H}$. Prove the following statements.

Find two closed linear subspaces $\mathcal{M}, \mathcal{N}$ of an infinite dimensional Hilbert space $\mathcal{H}$ such that $\mathcal{M} \cap \mathcal{N} = (0)$ and $\mathcal{M}+\mathcal{N}$ is dense in $\mathcal{H}$, but $\mathcal{M}+\mathcal{N} \neq \mathcal{H}$.

Define $A: l^{2}(\mathbb {Z}) \rightarrow l^{2}(\mathbb {Z})$ by $A(\ldots , \alpha_{-1}, \hat{\alpha }_{0}, \alpha_{1}, \ldots ) = (\ldots , \hat{\alpha }_{-1}, \alpha_{0}, \alpha_{1}, \ldots )$, where $\hat{\ }$ sits above the coefficient in the $0$-place. Find an invariant subspace of $A$ that does not reduce $A$. This operator is called the bilateral shift.

Let $\mu =$ Area measure on $\mathbb {D} \equiv \left\{ z \in \mathbb {C}: \left|z\right|<1\right\}$ and define $A: L^{2}(\mu ) \rightarrow L^{2}(\mu )$ by $(Af)(z) = zf(z)$ for $\left|z\right|<1$ and $f$ in $L^{2}(\mu )$. Find a nontrivial reducing subspace for $A$ and an invariant subspace that does not reduce $A$.