Some Applications of the Spectral Theorem

If $N$ is a normal operator show that $\operatorname {ran} N$ is closed if and only if $0$ is not a limit point of $\sigma (N)$.

Give an example of a non-normal operator $A$ such that $0$ is an isolated point of $\sigma (A)$ and $\operatorname {ran} A$ is closed. Give an example of a non-normal operator $B$ such that $\operatorname {ran} B$ is closed and $0$ is not an isolated point of $\sigma (B)$.

If $\mathcal{H}$ is a nonseparable Hilbert space find an example of a nontrivial closed ideal of $\mathcal{B}(\mathcal{H})$ that is different from $\mathcal{B}_0(\mathcal{H})$.

Let $(X, \Omega , \mu )$ be the measure space obtained in the proof of the following theorem (Theorem 4.6: if $N$ is a normal operator on $\mathcal{H}$, then there is a measure space $(X, \Omega , \mu )$ and a function $\phi$ in $L^\infty (X, \Omega , \mu )$ such that $N$ is unitarily equivalent to $M_\phi$ on $L^2(X, \Omega , \mu )$) and show that $L^1(X, \Omega , \mu )^*$ is isometrically isomorphic to $L^\infty (X, \Omega , \mu )$.

Show that $\mathcal{H}$ is separable if and only if every collection of pairwise orthogonal projections in $\mathcal{B}(\mathcal{H})$ is countable.

If $(X, \Omega , \mu )$ is a measure space, then $(X, \Omega , \mu )$ is $\sigma$-finite or $L^2(\mu )$ is finite dimensional if and only if every collection of pairwise orthogonal projections in $\{ M_\phi \in L^\infty (\mu )\}$ is countable.

If $N = \int z \, dE(z)$ and $\varepsilon > 0$, show that $\operatorname {ran} E(\{ z : \left|z\right| > \varepsilon \} ) \subseteq \operatorname {ran} N$.

(Calkin) Let $\mathcal{M}$ be a linear manifold in $\mathcal{H}$ and show that $\mathcal{M}$ has the property that $\mathcal{M}$ contains no closed infinite dimensional subspaces if and only if whenever $A \in \mathcal{B}(\mathcal{H})$ and $\operatorname {ran} A \subseteq \mathcal{M}$, then $A$ is compact.

Show that the extreme points of $\{ A \in \mathcal{B}(\mathcal{H}) : 0 \leq A \leq 1\}$ are the projections.

(Halmos) If $N$ is a normal operator, show that there is a hermitian operator $A$ and a continuous function $f$ such that $N = f(A)$. (Hint: Use the following result, Theorem 4.6. If $N$ is a normal operator on $\mathcal{H}$, then there is a measure space $(X, \Omega , \mu )$ and a function $\phi$ in $L^\infty (X, \Omega , \mu )$ such that $N$ is unitarily equivalent to $M_\phi$ on $L^2(X, \Omega , \mu )$.)