Math Problems Vault

Problem 11.1

(1)(2)(3)

Let $1 \leq p < +\infty$ . The operator $A \in \mathcal{L}\left(l_{p}\right)$ is defined by the formula $A x = \left(\alpha_{1} x_{1}, \alpha_{2} x_{2}, \ldots \right), x \in l_{p}$ , where $\left(\alpha_{1}, \alpha_{2}, \ldots \right) \in l_{\infty }$ . Prove that

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(1)

$\sigma_{p}(A) = \left\{ \alpha_{n}: n \geq 1\right\}$ ;

(2)

$\sigma (A) = \overline{\left\{ \alpha_{n}: n \geq 1\right\} }$ ;

(3)

$\sigma_{r}(A) = \varnothing$ .

Problem 11.2

(1)(2)

Let $H$ be a Hilbert space, $A \in \mathcal{L}(H)$ . Prove that

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(1)

$\sigma \left(A^{*}\right) = \left\{ \bar{\lambda } \mid \lambda \in \sigma (A)\right\}$ ;

(2)

$\left\{ \bar{\lambda } \mid \lambda \in \sigma_{r}(A)\right\} \subset \sigma_{p}\left(A^{*}\right) \subset \left\{ \bar{\lambda } \mid \lambda \in \sigma_{p}(A) \cup \sigma_{r}(A)\right\}$ .

Problem 11.3

Let $X = L_{2}(\left[a, b\right]),(A x)(t) = t x(t), t \in [a, b]$ . Find $\sigma (A), \sigma_{p}(A), \sigma_{c}(A)$ , and $\sigma_{r}(A)$ .

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Problem 11.4

Let $X = l_{2}$ . The operators $A, B \in \mathcal{L}\left(l_{2}\right)$ are defined as follows:

A x = \left(x_{2}, x_{3}, x_{4}, \ldots \right), B x = \left(0, x_{1}, x_{2}, \ldots \right), x \in l_{2}.

Find $\sigma (A), \sigma_{p}(A), \sigma_{c}(A), \sigma_{r}(A)$ and do the same for the operator $B$ .

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Problem 11.5

Let $X = L_{2}(\mathbb {R}),(A x)(t) = x(t+1), t \in \mathbb {R}$ . Find $\sigma (A), \sigma_{p}(A), \sigma_{c}(A)$ , and $\sigma_{r}(A)$ .

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