Math Problems Vault

Problem 8.1

Prove Theorem 8.1.

Theorem 8.1 (F. Riesz, 1910): the norm of the operator $A^{\prime }$ equals the norm of the operator $A$ , i.e., the conjugation operation preserves the norm.

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Problem 8.2

(a)(b)

Let $X, Y, Z$ be normed spaces, $A, B \in \mathcal{B}(X, Y)$ , $C \in \mathcal{B}(Y, Z), \alpha , \beta \in \mathbb {C}$ . Prove that

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(a)

$(\alpha A+\beta B)^{\prime } = \alpha A^{\prime }+\beta B^{\prime }$ ; $\quad$

(b)

$(C A)^{\prime } = A^{\prime } C^{\prime }$ .

Problem 8.3

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{B}(X, Y)$ , and let a sequence $g_{n} \stackrel{*}{\rightharpoonup } g$ in $Y^{*}$ . Prove that $A^{\prime } g_{n} \stackrel{*}{\rightharpoonup } A^{\prime } g$ in $X^{*}$ .

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Problem 8.4

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

Find the adjoint operators in the following cases.

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(a)

$X$ is an arbitrary normed space, $A = \lambda I, \lambda \in \mathbb {C}$ .

(b)

$X$ is a finite-dimensional space over the field $\mathbb {C}$ , the operator $A$ is written as a matrix $M$ in some linear basis. Let the same basis be chosen in the space $X^{*} = X$ . Find the matrix of the operator $A^{\prime }$ .

(c)

$X$ and $Y$ are Banach spaces, $X \subset Y$ as a linear subspace, $\left\| \cdot \right\|_{Y} \prec \left\| \cdot \right\|_{X}$ on $X, J \in \mathcal{B}(X, Y)$ is the embedding operator, i.e., $J: x \mapsto x$ ;

(d)

$X = X_{0} \oplus X_{1}$ is a Banach space, $X_{0}, X_{1}$ are its closed subspaces, $P \in \mathcal{B}(X)$ is the operator of projection onto $X_{0}$ along $X_{1}$ .

Problem 8.5

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

Find the adjoint operator $A^{\prime }$ for the operator $A$ in the spaces $l_{p}, p \in [1, \infty )$ , and $c_{0}$ :

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(a)

$A\left(x_{1}, x_{2}, \ldots \right) = \left(x_{2}, x_{3}, \ldots \right)$ , i.e., $A = T_{l}$ is the left shift;

(b)

$A\left(x_{1}, x_{2}, \ldots \right) = \left(0, x_{1}, x_{2}, \ldots \right)$ , i.e., $A = T_{r}$ is the right shift;

(c)

$A\left(x_{1}, x_{2}, \ldots \right) = \left(\lambda_{1} x_{1}, \lambda_{2} x_{2}, \ldots \right),\left\{ \lambda_{n}\right\}_{n = 1}^{\infty } \in l_{\infty }$ ;

(d)

$A\left(x_{1}, x_{2}, \ldots \right) = \left(x_{2}, \frac{x_{3}}{2}, \ldots , \frac{x_{n+1}}{n}, \ldots \right)$ .

Problem 8.6

Find the adjoint operator $A^{\prime }$ for the following operators $A$ :

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(a)

$A \in \mathcal{B}(C[0,1]),(A x)(t) = a(t) x(t), a \in C[0,1] ;$

(b)

$A \in \mathcal{B}\left(L_{p}[0,1]\right), p \in [1, \infty ),(A x)(t) = a(t) x(t), a \in L_{\infty }[0,1]$ ;

(c)

$A \in \mathcal{B}\left(L_{p}[0,1]\right), p \in [1, \infty ),(A x)(t) = \int_{0}^{t} x(s) d s$ ;

(d)

$A \in \mathcal{B}\left(L_{p}[0,1]\right), p \in [1, \infty )$ ,

(A x)(t) = \begin{cases} x(t+a), & \text{ if } t+a \leqslant 1 \\ x(t+a-1), & \text{ if } t+a > 1 \end{cases}

where $a \in [0,1]$ ;

(e)

$A \in \mathcal{B}(C[0,1]),(A x)(t) = \int_{0}^{1} K(t, s) x(s) d s$ , where $K \in C\left([0,1]^{2}\right)$ ;

(f)

$A \in \mathcal{B}(C[0,2])$ ,

(A x)(t) = \begin{cases} x(t), & \text{ if } t \in [0,1], \\ x(1), & \text{ if } t \in [1,2] ; \end{cases}

(g)

$A \in \mathcal{B}\left(C^{1}[0,1], C[0,1]\right),(A x)(t) = x^{\prime }(t)$ ;

(h)

$A \in \mathcal{B}(C[0,1]),(A x)(t) = x(0)+t x(1)+t^{2} \int_{0}^{1} x(s) d s$ .

Problem 8.7

Find the adjoint operator $A^{\prime }$ for the operator $A$ in the space $C[a, b]:(A x)(t) = \sum_{i = 1}^{n} \alpha_{i}(t) f_{i}(x)$ , where $\alpha_{i} \in C[a, b]$ , $f_{i} \in (C[a, b])^{*}$ , $f_{i}(x) = \int_{a}^{b} x(t) d w_{i}(t), w_{i} \in B V_{0}[0,1]$ .

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Problem 8.8

Let the operator $S$ in $c_{0}^{*} = l_{1}$ be given by the formula $S:\left(x_{1}, x_{2}, x_{3}, \ldots \right) \mapsto \left(\sum_{k = 1}^{\infty } x_{k}, x_{2}, x_{3}, \ldots \right)$ . Prove that $S$ is a linear isomorphism in $l_{1}$ , but is not the adjoint of any operator $T \in \mathcal{B}\left(c_{0}\right)$ .

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Problem 8.9

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{B}(X, Y)$ . Denote the canonical embeddings $\pi_{X}: X \hookrightarrow X^{* *}, \pi_{Y}: Y \hookrightarrow Y^{* *}$ . Prove that $A^{\prime \prime } \circ \pi_{X} = \pi_{Y} \circ A$ .

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Problem 8.10

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

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{B}(X, Y)$ . Prove that

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(a)

if $\overline{\operatorname {Im} A} = Y$ , then $\operatorname {Ker} A^{\prime } = \left\{ 0\right\}$ ;

(b)

if $\overline{\operatorname {Ker} A^{\prime }} = \left\{ 0\right\}$ , then $\overline{\operatorname {Im} A} = Y$ ;

(c)

if $\overline{\operatorname {Im} A^{\prime }} = X^{*}$ , then $\operatorname {Ker} A = \left\{ 0\right\}$ ;

(d)

if $X$ is reflexive and $\operatorname {Ker} A = \left\{ 0\right\}$ , then $\overline{\operatorname {Im} A^{\prime }} = X^{*}$ . Give an example of a non-reflexive space $X$ and an operator $A \in \mathcal{B}(X)$ with $\operatorname {Ker} A = \left\{ 0\right\}$ and $\overline{\operatorname {Im} A^{\prime }} \neq X^{*}$ .

Problem 8.11

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{B}(X, Y)$ . Prove that $(\operatorname {Im} A)^{\perp } = \operatorname {Ker} A^{\prime }$ , and $(\operatorname {Ker} A)^{\perp } \supseteq \overline{\operatorname {Im} A^{\prime }}$ (recall that for an arbitrary set $M \subset X$ the set $M^{\perp } = \left\{ f \in X^{*}: f(x) = 0 \text{ for every } x \in M\right\}$ ). Give an example where the last inclusion is strict.

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Problem 8.12

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{B}(X, Y)$ . Prove that if $A$ is a surjection, then $(\operatorname {Ker} A)^{\perp } = \operatorname {Im} A^{\prime }$ .

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Problem 8.13

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{B}(X, Y)$ . Prove that $\left(\operatorname {Im} A^{\prime }\right)^{\perp } = \operatorname {Ker} A,\left(\operatorname {Ker} A^{\prime }\right)^{\perp } = \overline{\operatorname {Im} A}$ (here for a set $M \subset X^{*}$ we denote $M^{\perp } = \left\{ x \in X: f(x) = 0\text{ for every }f \in M\right\}$ ).

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Problem 8.14

Let $X$ and $Y$ be Banach spaces, $A_{n} \xrightarrow {s} A$ in $\mathcal{B}(X, Y)$ . Does the sequence $\left\{ A_{n}^{\prime }\right\}_{n = 1}^{\infty }$ converge strongly to $A^{\prime }$ in $\mathcal{B}\left(Y^{*}, X^{*}\right)$ ? Does it converge weakly? Answer the same questions under the assumption that $X$ or $Y$ is reflexive.

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