Spectra of Banach-space adjoints

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Problem 11.27 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 11.2 "Problems to Solve"

(1)(2)(3)(4)

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

(1)

$\sigma \left(A^{\prime }\right) = \sigma (A)$ ;

(2)

$\sigma_{r}(A) \subset \sigma_{p}\left(A^{\prime }\right)$ ;

(3)

$\sigma_{p}(A) \subset \left(\sigma_{p}\left(A^{\prime }\right) \cup \sigma_{r}\left(A^{\prime }\right)\right)$ ;

(4)

$\sigma_{p}\left(A^{\prime }\right) \subset \left(\sigma_{p}(A) \cup \sigma_{r}(A)\right)$ .

Remark.

The inclusion $\sigma_{r}\left(A^{\prime }\right) \subset \sigma_{p}(A)$ is not always true, see problem 11.28, item (2).

Problem 12.13 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 12.1 "Spectrum"

(R. S. Phillips, 1955). Let $X$ be a Banach space, $A \in \mathcal{B}(X)$ . Prove that $\sigma (A) = \sigma \left(A^{\prime }\right)$ .

Problem 12.15 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 12.1 "Spectrum"

Let $X$ be a Banach space, $A \in \mathcal{B}(X)$ . Prove that if $\lambda \in \sigma_{\mathrm{r}}(A)$ , then $\lambda \in \sigma_{\mathrm{p}}\left(A^{\prime }\right)$ ; if $\lambda \in \sigma_{\mathrm{p}}(A)$ , then $\lambda \in \sigma_{\mathrm{p}}\left(A^{\prime }\right) \cup \sigma_{\mathrm{r}}\left(A^{\prime }\right)$ ; if $\lambda \in \sigma_{\mathrm{c}}(A)$ , then $\lambda \in \sigma_{\mathrm{c}}\left(A^{\prime }\right) \cup \sigma_{\mathrm{r}}\left(A^{\prime }\right)$ .

Problem 12.16 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 12.1 "Spectrum"

Let $X$ be a reflexive Banach space, $A \in \mathcal{B}(X)$ . Prove that $\sigma_{\mathrm{c}}(A) = \sigma_{\mathrm{c}}\left(A^{\prime }\right), \sigma_{\mathrm{r}}\left(A^{\prime }\right) \subset \sigma_{\mathrm{p}}(A)$ .

Problem VII.6.1 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section VII.6 "The Spectrum of a Linear Operator"

(a)(b)

Prove the following proposition.

(a)

If $\mathcal{X}$ is a Banach space and $A \in \mathcal{B}(\mathcal{X})$ , then $\sigma (A^*) = \sigma (A)$ .

(b)

If $\mathcal{H}$ is a Hilbert space and $A \in \mathcal{B}(\mathcal{H})$ , then $\sigma (A^*) = \sigma (A)^*$ , where for any subset $\Delta$ of $\mathbb {C}$ , $\Delta^* \equiv \left\{ \bar{z} : z \in \Delta \right\}$ .

Problem VII.6.2 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section VII.6 "The Spectrum of a Linear Operator"

Show that for $\mathcal{X}$ a Banach space and $A$ in $\mathcal{B}(\mathcal{X})$ , $\sigma_l(A) = \sigma_r(A^*)$ , where $\sigma_l$ and $\sigma_r$ denote the left and right spectra of $A$ . What happens in a Hilbert space?

References

[BCK+24] Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. Functional Analysis and Operator Theory. Springer, 2024.

[БСШ17] Бородин П.А., Савчук А.М., Шейпак И.А. Задачи по функциональному анализу. МЦНМО, 2017.

[Con07] Conway J.B. A Course in Functional Analysis. 2nd ed., Springer, 2007.