Math Problems Vault

Problem IX.5.1

Let $\mathcal{A}$ be a subalgebra of $\mathcal{B}(\mathcal{H})$ containing $1$ . Recall that for $1 \leq n \leq \infty$ , $\mathcal{H}^{(n)}$ denotes the direct sum of $\mathcal{H}$ with itself $n$ times, and if $A \in \mathcal{B}(\mathcal{H})$ , $A^{(n)}$ is the operator on $\mathcal{H}^{(n)}$ defined by $A^{(n)}(h_1, \dots , h_n) = (Ah_1, \dots , Ah_n)$ ; if $\mathcal{S} \subseteq \mathcal{B}(\mathcal{H})$ , $\mathcal{S}^{(n)} \equiv \{ A^{(n)} : A \in \mathcal{S}\}$ . Proposition 5.3 states that the SOT closure of $\mathcal{A}$ is

\{ B \in \mathcal{B}(\mathcal{H}) : \text{for every finite } n, \ \operatorname {Lat} \mathcal{A}^{(n)} \subseteq \operatorname {Lat} B^{(n)}\} .

Show that if $B \in \text{SOT-}\operatorname {cl} \mathcal{A}$ , then $B$ belongs to the set defined above.

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Problem IX.5.2

Show that $\mathcal{B}_{00}$ is SOT dense in $\mathcal{B}$ .

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Problem IX.5.3

If $\{ A_k\}$ and $\{ B_k\}$ are sequences in $\mathcal{B}(\mathcal{H})$ such that $A_k \to A$ (WOT) and $B_k \to B$ (SOT), then $A_k B_k \to AB$ (WOT).

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Problem IX.5.4

With the notation of Exercise 3, show that if $A_k \to A$ (SOT), then $A_k B_k \to AB$ (SOT).

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Problem IX.5.5

Let $S$ be the unilateral shift on $l^2(\mathbb {N})$ (II.2.10). Examine the sequences $\{ S^k\}$ and $\{ S^{*k}\}$ and their relation to Exercises 3 and 4.

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Problem IX.5.6

(a)(b)

(Halmos) Fix an orthonormal basis $\{ e_n : n \geq 1\}$ for $\mathcal{H}$ .

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(a)

Show that $0 \in$ weak closure of $\{ \sqrt{n} e_n : n \geq 1\}$ (Halmos [1982], Solution 28).

(b)

Let $\{ n_i\}$ be a net of integers such that $\sqrt{n_i} e_{n_i} \to 0$ weakly. Define $A_i f = \sqrt{n_i} \left\langle f, e_{n_i} \right\rangle e_{n_i}$ for $f$ in $\mathcal{H}$ . Show that $A_i \to 0$ (SOT) but $\{ A_i^2\}$ does not converge to $0$ (SOT).