Math Problems Vault

Problem IX.6.1

If $\mathcal{S} \subseteq \mathcal{B}(\mathcal{H})$ , show that $\mathcal{S}' = \mathcal{S}'''$ .

?

Problem IX.6.2

If $\mathcal{S} \subseteq \mathcal{B}(\mathcal{H})$ , show that $\mathcal{S}'$ is always a SOT closed subalgebra of $\mathcal{B}(\mathcal{H})$ .

?

Problem IX.6.3

(a)(b)

Prove the following proposition (Proposition 6.1).

?

(a)

If $A = A_1 \oplus A_2 \oplus \cdots$ is a bounded operator on $\mathcal{H} = \mathcal{H}_1 \oplus \mathcal{H}_2 \oplus \cdots$ and $B = [B_{ij}] \in \mathcal{B}(\mathcal{H})$ , then $AB = BA$ if and only if $B_{ij} A_j = A_i B_{ij}$ for all $i, j$ .

(b)

If $B = [B_{ij}] \in \mathcal{B}(\mathcal{H}^{(n)})$ , $BA^{(n)} = A^{(n)} B$ if and only if $B_{ij} A = A B_{ij}$ for all $i, j$ .

Problem IX.6.4

(a)(b)

Let $\mathcal{H}$ be a Hilbert space of dimension $\alpha$ and define $S: \mathcal{H}^{(\infty )} \to \mathcal{H}^{(\infty )}$ by $S(h_1, h_2, \dots ) = (0, h_1, h_2, \dots )$ . $S$ is called the unilateral shift of multiplicity $\alpha$ .

?

(a)

Show that $A = [A_{ij}] \in \{ S\} '$ if and only if $A_{ij} = 0$ for $j > i$ and $A_{ij} = A_{i+1, j+1}$ for $i \geq j$ .

(b)

Show that $A = [A_{ij}] \in \{ S\} ''$ if and only if $A_{ij} = 0$ for $j > i$ and $A_{ij} = A_{i+1, j+1} =$ a multiple of the identity for $i \geq j$ .

Problem IX.6.5

What is $\{ N_\mu \oplus N_\mu \} '$ ? $\{ N_\mu \oplus N_\mu \} ''$ ?

?

Problem IX.6.6

If $\mathcal{A}$ is a subalgebra of $\mathcal{B}(\mathcal{H})$ , show that $\mathcal{A}$ is a maximal abelian subalgebra of $\mathcal{B}(\mathcal{H})$ if and only if $\mathcal{A} = \mathcal{A}'$ .

?

Problem IX.6.7

Find a non-normal operator that is similar to a normal operator. (Hint: Try $\dim \mathcal{H} = 2$ .)

?

Problem IX.6.8

Let $\mu$ be a compactly supported measure on $\mathbb {C}$ and let $\mathcal{X}$ be a separable Hilbert space. A function $f: \mathbb {C} \to \mathcal{X}$ is a Borel function if $f^{-1}(G)$ is a Borel set when $G$ is weakly open in $\mathcal{X}$ . Define $L^2(\mu , \mathcal{X})$ to be the equivalence classes of Borel functions $f: \mathbb {C} \to \mathcal{X}$ such that $\int \left\| f(x)\right\|^2 \, d\mu (x) < \infty$ . Define $\left\langle f, g \right\rangle = \int \left\langle f(x), g(x) \right\rangle \, d\mu (x)$ for $f$ and $g$ in $L^2(\mu , \mathcal{X})$ .

?

(a)

Show that $L^2(\mu , \mathcal{X})$ is a Hilbert space. Define $N$ on $L^2(\mu , \mathcal{X})$ by $(Nf)(z) = zf(z)$ .

(b)

Show that $N$ is a normal operator and $\sigma (N) =$ support $\mu$ . Calculate $N^*$ .

(c)

Show that $N \cong N_\mu^{(\alpha )}$ , where $\alpha = \dim \mathcal{X}$ .

(d)

Find $\{ N\} '$ . (Hint: Use (6.1).)

(e)

Find $\{ N\} ''$ .

Problem IX.6.9

Let $\mathcal{H}$ be separable with basis $\{ e_n\}$ . Let $A$ be the diagonal operator on $\mathcal{H}$ given by $Ae_n = \lambda_n e_n$ , where $\sup_n \left|\lambda_n\right| < \infty$ . Determine $\{ A\} '$ and $\{ A\} ''$ . Give necessary and sufficient conditions on $\{ \lambda_n\}$ such that $\{ A\} ' = \{ A\} ''$ .

?

Problem IX.6.10

Let $\mathcal{A}$ be a $C^*$ -subalgebra of $\mathcal{B}(\mathcal{H})$ but do not assume that $\mathcal{A}$ contains the identity operator. Let $\mathcal{M} = \bigvee \{ \operatorname {ran} A : A \in \mathcal{A}\}$ and let $P =$ the projection of $\mathcal{H}$ onto $\mathcal{M}$ . Show that $\text{SOT-}\operatorname {cl} \mathcal{A} = \mathcal{A}'' P = P \mathcal{A}''$ .

?

Problem IX.6.11

Formulate and prove a polar decomposition for operators between different Hilbert spaces.

?

Problem IX.6.12

(a)(b)(c)

Let $(X, \Omega , \mu )$ be an arbitrary measure space and let $L \in L^1(\mu )^*$ .

?

(a)

Show that for every $f$ in $L^2(\mu )$ , there is an $h$ in $L^2(\mu )$ such that $L(g\bar{f}) = \int g \bar{h} \, d\mu$ for all $g$ in $L^2(\mu )$ .

(b)

If $f \in L^2(\mu )$ , let $Tf$ be the function $h$ in $L^2(\mu )$ obtained in part (a). Show that $T: L^2(\mu ) \to L^2(\mu )$ defines a bounded linear operator and $T$ commutes with $M_\phi$ for every $\phi$ in $L^\infty (\mu )$ .

(c)

In light of parts (a) and (b), compare Theorem 6.6 and Example 20.17 in Hewitt and Stromberg [1975].