Math Problems Vault

Problem XI.1.1

Let $S$ be the unilateral shift of multiplicity $1$ on $l^2(\mathbb {N})$ and find $\sigma_l(S)$ and $\sigma_r(S)$ .

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Problem XI.1.2

(a)(b)(c)(d)

The compression spectrum of $A$ , $\sigma_c(A)$ , is defined by $\sigma_c(A) = \left\{ \lambda \in \mathbb {C}: \mathrm{ran}(A - \lambda ) \text{ is not dense in } \mathcal{H}\right\}$ . Show:

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(a)

$\lambda \in \sigma_c(A)$ if and only if $\bar{\lambda } \in \sigma_p(A^*)$ .

(b)

$\sigma_c(A) \subseteq \sigma_r(A)$ , but this inclusion may be proper.

(c)

$\sigma_c(A)$ is not necessarily closed.

(d)

$\sigma (A) = \sigma_{ap}(A) \cup \sigma_c(A)$ .

Problem XI.1.3

If $A \in \mathcal{B}(\mathcal{H})$ and $f \in \mathrm{Hol}(A)$ , then $f(\sigma_p(A)) \subseteq \sigma_p(f(A))$ . If $f$ is not constant on any component of its domain, then $f(\sigma_p(A)) = \sigma_p(f(A))$ .

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Problem XI.1.4

If $A \in \mathcal{B}(\mathcal{H})$ and $f \in \mathrm{Hol}(A)$ , then $f(\sigma_{ap}(A)) = \sigma_{ap}(f(A))$ .

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