Unitary Equivalence for Compact Normal Operators

Show that ``unitary equivalence'' is an equivalence relation on $\mathcal{B}(\mathcal{H})$.

Let $U: \mathcal{H} \rightarrow \mathcal{K}$ be an isomorphism and define $\rho : \mathcal{B}(\mathcal{H}) \rightarrow \mathcal{B}(\mathcal{K})$ by $\rho (A) = UAU^{-1}$. Prove:

Say that an operator $A$ on $\mathcal{H}$ is irreducible if the only reducing subspaces for $A$ are $(0)$ and $\mathcal{H}$. Prove:

Suppose $A = \oplus \left\{ A_{i}: i \in I\right\}$ and $B = \oplus \left\{ B_{i}: i \in I\right\}$ where each $A_{i}$ and $B_{i}$ is irreducible (Exercise 3). Show that $A \cong B$ if and only if there is a bijection $\pi : I \rightarrow I$ such that $A_{i} \cong B_{\pi (i)}$.

If $T$ is a compact normal operator and $m_{T} = m$ is its multiplicity function, prove:

Show that two projections $P$ and $Q$ are unitarily equivalent if and only if $\dim (\operatorname {ran} P) = \dim (\operatorname {ran} Q)$ and $\dim (\ker P) = \dim (\ker Q)$.

Let $A: L^{2}(0,1) \rightarrow L^{2}(0,1)$ be defined by $(Af)(x) = xf(x)$ for $f$ in $L^{2}(0,1)$ and $x$ in $(0,1)$. Show that $A \cong A^{2}$.

Say that a compact normal operator $T$ is simple if $m_{T} \leq 1$. (See Exercises 7.10 and 7.11.) Show that every compact normal operator $T$ on a separable Hilbert space is unitarily equivalent to $\oplus_{n=1}^{\infty } T_{n}$, where each $T_{n}$ is a simple compact normal operator and $m_{T_{n}} \geq m_{T_{n+1}}$ for all $n$. Show that $\left\| T_{n}\right\| \rightarrow 0$. (Of course, there may only be a finite number of $T_{n}$.)

Using the notation of Exercise 8, suppose also that $S$ is a compact normal operator and $S \cong \oplus_{n=1}^{\infty } S_{n}$, where $S_{n}$ is a simple compact normal operator and $m_{S_{n}} \geq m_{S_{n+1}}$ for all $n$. Show that $T \cong S$ if and only if $T_{n} \cong S_{n}$ for all $n$.

If $T$ is a compact normal operator on a separable Hilbert space, show that there are simple compact normal operators $T_{1}, T_{2}, \ldots$ such that $T \cong 0 \oplus T_{1} \oplus T_{2}^{(2)} \oplus T_{2}^{(3)} \oplus \cdots$, where:

Using the notation of Exercise 10, let $S$ be a compact normal operator and let $0 \oplus S_{1} \oplus S_{2}^{(2)} \oplus \cdots$ be the corresponding decomposition. Show that $T \cong S$ if and only if $T_{n} \cong S_{n}$ and $\ker T$ and $\ker S$ have the same dimension.

If $T$ is a non-zero compact normal operator, show that $T$ and $T \oplus T$ are not unitarily equivalent.

Give an example of a nontrivial operator $T$ such that $T \cong T \oplus T$. Show that if $T \cong T \oplus T$, then $T \cong T \oplus T \oplus \cdots$. Characterize the diagonalizable normal operators $T$ such that $T \cong T \oplus T$.

Let $\mathcal{H}$ be the space defined in Example I.1.8 and let $U: \mathcal{H} \rightarrow L^{2}(0,1)$ be the isomorphism defined by $Uf = f'$ (Exercise I.1.4). If $(Af)(x) = xf(x)$ for $f$ in $\mathcal{H}$, what is $UAU^{-1}$?

Example I.1.8 (context): Let $\mathcal{H}$ = the collection of all absolutely continuous functions $f: [0,1] \rightarrow \mathbb {F}$ such that $f(0)=0$ and $f' \in L^{2}(0,1)$. If $\left\langle f, g \right\rangle = \int_{0}^{1} f'(t) \overline{g'(t)} \, dt$ for $f$ and $g$ in $\mathcal{H}$, then $\mathcal{H}$ is a Hilbert space.