Math Problems Vault

Problem XI.4.1

Show that the material of this section is only significant for infinite dimensional Hilbert spaces by showing that the essential spectrum of every operator on $\mathcal{H}$ is non-empty if and only if $\mathcal{H}$ is infinite dimensional.

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Problem XI.4.2

Let $G$ be a bounded region in $\mathbb {C}$ such that $\partial G = \partial [\mathrm{cl}\ G]$ and let $\phi$ be a function that is analytic in a neighborhood of $\mathrm{cl}\ G$ . Define $A: L_a^2(G) \to L_a^2(G)$ by $Af = \phi f$ . Find all of the parts of the spectrum of $A$ .

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Problem XI.4.3

(Fillmore, Stampfli, and Williams [1972].) If $\lambda \in \sigma_{le}(A)$ , then there is a projection $P$ , having infinite rank, such that $\pi (A - \lambda )\pi (P) = 0$ .

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Problem XI.4.4

(a)(b)(c)

(Fillmore, Stampfli, and Williams [1972].) Let $A \in \mathcal{B}(\mathcal{H})$ .

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(a)

If $A$ has a cyclic vector $e$ , show that $\dim \left\{ Ae, A^2e, \ldots \right\}^\perp \le 1$ .

(b)

Let $\lambda \in \sigma_{le}(A^*)$ . If $\varepsilon > 0$ , let $f_1, f_2$ be orthonormal vectors such that $\left\| (A^* - \lambda )f_j\right\| < \varepsilon$ for $j = 1, 2$ and let $P =$ the projection onto $\bigvee \left\{ f_1, f_2\right\}$ . Put $B = \bar{\lambda }P + (1 - P)A$ . Show that $\left\| B - A\right\| < 2\varepsilon$ .

(c)

Show that the noncyclic operators are dense in $\mathcal{B}(\mathcal{H})$ if $\dim \mathcal{H} > 1$ .

Problem XI.4.5

Let $A \in \mathcal{F}$ and suppose $f$ is analytic in a neighborhood of $\sigma (A)$ and does not vanish on $\sigma_e(A)$ . Show that $f(A) \in \mathcal{F}$ and find $\operatorname {ind} f(A)$ .

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Problem XI.4.6

Let $S$ be the unilateral shift and let $f$ be an analytic function in a neighborhood of $\mathrm{cl}\ \mathbb {D}$ such that $f(z) \ne 0$ if $\left|z\right| = 1$ . Let $\gamma (t) = f(\exp (2\pi i t))$ , $0 \le t \le 1$ . Show that $\sigma_e(f(S)) = f(\partial \mathbb {D}) = \left\{ \gamma (t): 0 \le t \le 1\right\}$ and that if $\lambda \notin f(\partial \mathbb {D})$ , $\operatorname {ind}(f(S) - \lambda ) = -n(\gamma ; \lambda )$ , where $n(\gamma ; \lambda ) =$ the winding number of $\gamma$ about $\lambda$ . Moreover, show that if $\operatorname {ind}(f(S) - \lambda ) = 0$ , then $\lambda \notin \sigma (f(S))$ .

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Problem XI.4.7

Let $S$ be the operator defined in Example 2.11 where $G = \mathbb {D}$ . Show that there is a compact operator $K$ such that $S + K$ is unitarily equivalent to the unilateral shift.

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Problem XI.4.8

If $S$ is the unilateral shift, show that for every $\varepsilon > 0$ there is a rank one operator $F$ with $\left\| F\right\| < \varepsilon$ such that $\sigma (S^* \oplus S + F) = \partial \mathbb {D}$ .

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Problem XI.4.9

Let $G$ be an open connected subset of $\sigma (A) \setminus \sigma_{le}(A) \cup \sigma_{re}(A)$ and suppose $\lambda_0 \in G$ such that $\operatorname {ind}(A - \lambda_0) = 0$ . Show that there is a finite rank operator $F$ such that $A + F - \lambda_0$ is invertible. Show that $A + F - \lambda$ is invertible for every $\lambda$ in $G$ .

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