Schauder's theorem on compact adjoints

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Problem 10.74 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 10.2 "Problems to Solve"

(Schauder) Let $A \in \mathcal{L}\left(X_{1}, X_{2}\right)$ , where $X_{1}$ is an NLS and $X_{2}$ is a Banach space. Prove that $A \in S_{\infty }\left(X_{1}, X_{2}\right)$ if and only if $A^{\prime } \in S_{\infty }\left(X_{2}^{*}, X_{1}^{*}\right)$ .

Problem 10.78 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 10.2 "Problems to Solve"

Let $A \in \mathcal{L}\left(X_{1}, X_{2}\right)$ , where $X_{1}$ is an NLS and $X_{2}$ is a separable Banach space. Prove that $A \in S_{\infty }\left(X_{1}, X_{2}\right)$ if and only if $A^{\prime }$ transforms every weak-* convergent sequence in $X_{2}^{*}$ into a strongly convergent sequence in $X_{1}^{*}$ .

Problem 11.24 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 11.1 "General properties of compact operators"

Let $X$ and $Y$ be Banach spaces, with $Y$ separable. Let $A \in \mathcal{B}(X, Y)$ , and let the operator $A^{\prime }$ carry every $*$ -weakly convergent sequence in $Y^{*}$ into a sequence convergent in norm in $X^{*}$ . Prove that $A$ is compact.

Problem 11.50 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 11.3 "Compact operators in Hilbert spaces"

Prove that in a Hilbert space the operator $A$ is compact if and only if $A^{*}$ is compact.

Problem B.7 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Appendix B "Compact and Hilbert–Schmidt Operators"

(a)(b)

(a)

Given $x_{k} \in H$ and $y_{k} \in K$ , show that the adjoint of $T = \sum_{k = 1}^{M} x_{k} \otimes y_{k}$ is $T^{*} = \sum_{k = 1}^{M} y_{k} \otimes x_{k}$

(b)

Given $T \in \mathcal{B}(H, K)$ , show that $T$ is compact if and only if $T^{*}$ is compact.

Problem 343 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection III.3.2 "Compact Sets and Compact Operators"

Let $L_1$ and $L_2$ be Banach spaces, and $A\in \mathcal L(L_1,L_2)$ . Prove that $A$ is compact if $A'$ is.

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.

[Hei11] Heil C. A Basis Theory Primer. Birkhäuser, 2011.

[KiGv88] Kirillov A.A., Gvishiani A.D. Теоремы и задачи функционального анализа. 2-е изд., Наука, 1988.