Math Problems Vault

Problem 10.5

Let $M$ be a set in a Banach space such that for every $\varepsilon > 0$ there exists a precompact $\varepsilon$ -net for the set $M$ . Prove that the set $M$ is precompact.

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Problem 10.6

Prove that every subset of a precompact set is precompact.

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Problem 10.7

Prove that if at least one (open or closed) ball in an NLS is precompact, then all open and closed balls in this space are precompact.

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Problem 10.8

Let $M$ be a precompact set in a metric space $(X, \rho )$ . A sequence $\left\{ x_{n}: n \geq \right\}$ 1) $\subset X$ is such that $\rho \left(x_{n}, M\right) \rightarrow 0, n \rightarrow \infty$ . Prove that the set $K = \left\{ x_{n}: n \geq 1\right\}$ is precompact.

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Problem 10.9

(1)(2)

Let $M = \left\{ x_{n}: n \geq 1\right\}$ be a precompact set in an NLS $X, x_{0} \in X$ . Prove that $x_{n} \rightarrow x_{0}, n \rightarrow \infty$ , in $X$ , if at least one of the following conditions is fulfilled:

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(1)

$x_{0}$ is the only limit point of the set $M$ ;

(2)

$x_{n} \xrightarrow {w} x_{0}, n \rightarrow \infty$ .

Problem 10.10

Let $\left\{ M_{n}: n \geq 1\right\}$ be a sequence of compact sets in a metric space. Prove that if $\bigcap_{k = 1}^{n} M_{k} \neq \varnothing , n \geq 1$ , then $\bigcap_{k = 1}^{\infty } M_{k} \neq \varnothing$ .

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Problem 10.11

Let $M$ be a compact set in a metric space $(X, \rho )$ . Prove that for any $\varepsilon > 0$ the set $M$ can be represented as $M = \bigcup_{k = 1}^{n} M_{k}$ , where $M_{k}$ are closed sets for which

\operatorname {diam} M_{k} = \sup _{x, y \in M_{k}} \rho (x, y) \leq \varepsilon , 1 \leq k \leq n.

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Problem 10.12

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

Let $X$ be an NLS, $M \subset X, N \subset X$ . Is it correct that the set $M+N$ :

?

(1)

is compact, if $M$ and $N$ are compact;

(2)

is precompact, if $M$ and $N$ are precompact;

(3)

is closed, if $M$ is compact and $N$ is closed;

(4)

is closed, if $M$ and $N$ are closed?

Problem 10.13

(1)(2)(3)

Let $(X, \rho )$ be a metric space, $M \subset X$ . A function $f: M \rightarrow \mathbb {R}$ is called lower semicontinuous (upper semicontinuous) on the set $M$ , if

\forall x_{0} \in M \forall \varepsilon > 0 \exists \delta > 0 \forall x \in B\left(x_{0}, \delta \right) \cap M: f(x) > f\left(x_{0}\right)-\varepsilon

(or $f(x) < f\left(x_{0}\right)+\varepsilon$ , respectively). Prove that

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(1)

if $M$ is a compact set, then every lower (upper) semicontinuous function on $M$ attains its minimum (maximum) value on $M$ ;

(2)

the converse statement to item (1) is also correct;

(3)

if every continuous function on $M$ is bounded, then $M$ is a compact set.

Problem 10.14

Let $M$ be a set in a Banach space such that every real continuous function on $M$ is uniformly continuous. Does it follow that $M$ is a compact set?

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Problem 10.15

Let $\left\{ L_{n}: n \geq 1\right\}$ be a sequence of finite-dimensional subspaces of a Banach space $X$ such that $L_{n} \subset L_{n+1}, n \geq 1$ , and the set $\bigcup_{n = 1}^{\infty } L_{n}$ is dense in $X$ . Prove that a bounded set $M$ is precompact in $X$ if and only if $\sup_{x \in M} \rho \left(x, L_{n}\right) \rightarrow 0, n \rightarrow \infty$ .

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Problem 10.16

(1)(2)(3)

Prove that a bounded set $M$ is precompact

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(1)

in the space $l_{p}, 1 \leq p < +\infty$ , if and only if

\forall \varepsilon > 0 \exists N = N(\varepsilon ) \in \mathbb {N} \forall x \in M: \sum _{k = N+1}^{\infty }\left|x_{k}\right|^{p} < \varepsilon ;

(2)

in the space $c_{0}$ if and only if

\forall \varepsilon > 0 \exists N = N(\varepsilon ) \in \mathbb {N} \forall x \in M \forall k \geq N:\left|x_{k}\right| < \varepsilon ;

(3)

in the space $c$ if and only if

\forall \varepsilon > 0 \exists N = N(\varepsilon ) \in \mathbb {N} \forall x \in M \forall k \geq N:\left|x_{k}-\lim _{n \rightarrow \infty } x_{n}\right| < \varepsilon .

Problem 10.17

Formulate and prove the criterion of precompactness in the separable Hilbert space $H$ with an orthonormal basis $\left\{ e_{n}: n \geq 1\right\}$ .

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Problem 10.18

Prove that the following sets are precompact in $C(\left[a, b\right])$ (here $k_{1}, k_{2}$ are fixed nonnegative constants):

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(1)

$\left\{ \int_{a}^{t} x(s) d s, t \in [a, b] \mid x \in M\right\}$ , where $M$ is a bounded set in $C(\left[a, b\right])$ ;

(2)

$\left\{ x \in C(\left[a, b\right]) \mid \left|x(a)\right| \leq k_{1}, \left|x\left(t_{1}\right)-x\left(t_{2}\right)\right| \leq k_{2}\left|t_{1}-t_{2}\right|, t_{1}, t_{2} \in [a, b]\right\}$ ;

(3)

$\left\{ x \in C^{1}(\left[a, b\right]) \mid \left|x\left(t_{0}\right)\right| \leq k_{1}, \left|x^{\prime }(t)\right| \leq k_{2}, t \in [a, b]\right\}$ , where $t_{0} \in [a, b]$ is fixed;

(4)

$\left\{ \int_{a}^{t} K(t, s) x(s) d u, t \in [a, b] \mid x \in B(0,1)\right\}$ , where $K \in C\left(\left[a, b\right]^{2}\right)$ ;

(5)

$\left\{ x \in C^{1}(\left[a, b\right]) \mid \left|x(a)\right| \leq k_{1}, \int_{a}^{b}\left|x^{\prime }(t)\right|^{2} d t \leq k_{2}\right\}$ ;

(6)

$\left\{ x \in C^{1}(\left[a, b\right]) \mid \int_{a}^{b}\left(\left|x(t)\right|^{p}+\left|x^{\prime }(t)\right|^{p}\right) d t \leq k_{1}\right\}$ , where $1 < p < +\infty$ ;

(7)

$\left\{ x \in C^{2}(\left[a, b\right]) \mid \left|x(t)\right| \leq k_{1}, \left|x^{\prime \prime }(t)\right| \leq k_{2}, t \in [a, b]\right\}$ ;

(8)

\left\{ x \in C^{1}(\left[a, b\right]) \mid \left|x(t)\right| \leq k_{1}, t \in [a, b], \left|x^{\prime }\left(t_{1}\right)-x^{\prime }\left(t_{2}\right)\right| \leq k_{2}\left|t_{1}-t_{2}\right|, t_{1}, t_{2} \in [a, b]\right\} ;

(9)

an equicontinuous set of continuous functions, whose absolute values are bounded by a fixed constant at some point $t_{0} \in [a, b]$ ;

(10)

a bounded set of polynomials of degree $n$ .

Which of these sets are compact in $C(\left[a, b\right])$ ?

Problem 10.19

Which of the following sets are precompact (compact) in $C(\left[0,1\right])$ :

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(1)

$\left\{ t^{n} \mid n \geq 1\right\}$ ;

(2)

$\left\{ \sin n \sqrt{t} \mid n \geq 1\right\}$ ;

(3)

${ }^{*}\left\{ \sin (t+n) \mid n \geq 1\right\}$ ;

(4)

$\left\{ \sin \alpha t \mid \alpha \in \mathbb {R}\right\}$ ;

(5)

$\left\{ \sin \alpha t \mid \alpha \in [1,3]\right\}$ ;

(6)

$\left\{ \arctan \alpha t \mid \alpha \in \mathbb {R}\right\}$ ;

(7)

$\left\{ e^{t-\alpha } \mid \alpha \in \mathbb {R}\right\}$ ;

(8)

$\left\{ e^{t-\alpha } \mid \alpha \geq 0\right\}$ ;

(9)

$\left\{ \sum_{n = 1}^{\infty } \frac{a_{n}}{n^{3}} e^{-n t} \mid a_{n} \in (-1,1), n \geq 1\right\}$ ;

(10)

$\left\{ \sum_{n = 1}^{\infty } \frac{a_{n}}{t+n^{2}} \mid a_{n} \in (-1,1), n \geq 1\right\}$ ?

Problem 10.20

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

Under what conditions on a set $A \subset \mathbb {R}$ are the following sets precompact (compact) in $C(\left[0,1\right])$ :

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(1)

$\left\{ \sin \alpha t \mid \alpha \in A\right\}$ ;

(2)

$\left\{ \arctan \alpha t \mid \alpha \in A\right\}$ ;

(3)

$\left\{ e^{t-\alpha } \mid \alpha \in A\right\}$ ;

(4)

$\left\{ \left.\frac{1}{1+\left|t-\alpha \right|} \right\rvert \, \alpha \in A\right\}$ ?

Problem 10.21

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

Are the following sets precompact in $L_{2}(\left[0,1\right])$ :

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(1)

$\left\{ t^{n} \mid n \geq 1\right\}$ ;

(2)

$\left\{ \sin \pi n t \mid n \geq 1\right\}$ ;

(3)

$\left\{ \sin \alpha t \mid \alpha \in \mathbb {R}\right\}$ ;

(4)

$\left\{ \arctan \alpha t \mid \alpha \in \mathbb {R}\right\}$ ?

Problem 10.22

(1)(2)

Prove the compactness of the following sets in $l_{2}$ :

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(1)

$\left\{ x \in l_{2} \mid \forall n \geq 1: \left|x_{n}\right| \leq \frac{1}{n}\right\}$ ;

(2)

$\left\{ x \in l_{2} \mid \sum_{n = 1}^{\infty } n^{2}\left|x_{n}\right|^{2} \leq 1\right\}$ .

Problem 10.23

(1)(2)

Let $a_{n} > 0, n \in \mathbb {N}$ . Formulate conditions on $\left\{ a_{n}: n \geq 1\right\}$ under which the following sets are compact in $l_{2}$ :

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(1)

$\left\{ x \in l_{2} \mid \forall n \geq 1: \left|x_{n}\right| \leq \frac{1}{a_{n}}\right\}$ ;

(2)

$\left\{ x \in l_{2} \mid \sum_{n = 1}^{\infty } a_{n}\left|x_{n}\right|^{2} \leq 1\right\}$ .

Problem 10.24

Prove that every set which is precompact in $C^{1}(\left[a, b\right])$ is also precompact in $C(\left[a, b\right])$ . Is it true that every subset of $C^{1}(\left[a, b\right])$ which is precompact in $C(\left[a, b\right])$ is also precompact in $C^{1}(\left[a, b\right])$ ?

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Problem 10.25

Prove that the ball $\bar{B}(0,1)$ of the space $C^{1}(\left[a, b\right])$ is precompact, but not a closed set in $C(\left[a, b\right])$ .

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Problem 10.26

Formulate and prove the criterion of precompactness in the spaces $C^{k}(\left[a, b\right]), k \in \mathbb {N}$ .

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Problem 10.27

Let $(K, \rho )$ be a compact metric space. Prove that a set $M$ is precompact in the space $C(K)$ if and only if it is uniformly bounded and equicontinuous (i.e.,

\forall \varepsilon > 0 \exists \delta > 0 \forall t_{1}, t_{2} \in K, \rho \left(t_{1}, t_{2}\right) < \delta , \forall x \in M:\left|x\left(t_{1}\right)-x\left(t_{2}\right)\right| < \varepsilon ).

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Problem 10.28

Let $X_{1}, X_{2}$ be normed linear spaces and $X_{1}$ is a subset of $X_{2}$ . Find the necessary and sufficient conditions for every precompact set in $X_{1}$ to be precompact in $X_{2}$ .

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Problem 10.29

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

Is it true that every precompact set in an NLS $X_{1}$ is precompact in an NLS $X_{2}$ , if

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(1)

$X_{1} = L_{2}(\left[a, b\right]), X_{2} = L_{1}(\left[a, b\right])$ ;

(2)

$X_{1} = C(\left[a, b\right]), X_{2} = L_{2}(\left[a, b\right])$ ;

(3)

$X_{1} = l_{1}, X_{2} = l_{\infty }$ ;

(4)

$X_{1} = l_{1}, X_{2} = l_{2}$ ?

Problem 10.30

(1)(2)

Let $T$ be some set, $B(T)$ be the linear space of all bounded functions from $T$ to $\mathbb {R}$ , and $\left\| x\right\|_{\infty } = \sup_{t \in T}\left|x(t)\right|, x \in B(T)$ . Prove the following statements.

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(1)

$B(T)$ with the norm $\left\| \cdot \right\|_{\infty }$ is a Banach space.

(2)

A set $M$ is precompact in the space $B(T)$ if and only if it is bounded and for every $\varepsilon > 0$ there exists a finite partition $T = \cup_{k = 1}^{n} T_{k}$ such that each function from $M$ on each subset $T_{k}$ varies at most by $\varepsilon$ , i.e.,

\forall 1 \leq k \leq n \forall x \in M \forall t_{1}, t_{2} \in T_{k}:\left|x\left(t_{1}\right)-x\left(t_{2}\right)\right| \leq \varepsilon .

Problem 10.31

Let $T$ be some set, $Y$ be a compact subset of an NLS $X,\left\| \cdot \right\|$ be a norm in $X, B(T, Y)$ be the linear space of all mappings from $T$ to $Y$ , and $\left\| x\right\|_{\infty } = \sup \left\| x(t)\right\| , x \in B(T, Y)$ . Prove that $B(T, Y)$ with the norm $\left\| \cdot \right\|_{\infty }$ is a Banach $t \in T$ space. Formulate and prove the criterion of precompactness in this space.

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Problem 10.32

Let $X_{1}, X_{2}$ be normed linear spaces. Prove that a linear operator $A: X_{1} \rightarrow X_{2}$ is compact if and only if for every bounded sequence $\left\{ x_{n}: n \geq 1\right\} \subset X_{1}$ the sequence $\left\{ A x_{n}: n \geq 1\right\} \subset X_{2}$ contains a convergent subsequence.

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Problem 10.33

(1)(2)

Prove that in an infinite-dimensional NLS

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(1)

the identity operator is not compact;

(2)

every compact operator is not continuously invertible.

Problem 10.34

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

Let $X_{1}, X_{2}, X_{3}$ be the normed linear spaces. Prove that

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(1)

$S_{0}\left(X_{1}, X_{2}\right) \subset S_{\infty }\left(X_{1}, X_{2}\right)$ ;

(2)

$S_{0}\left(X_{1}, X_{2}\right)$ is a linear set in $\mathcal{L}\left(X_{1}, X_{2}\right)$ ;

(3)

if $A \in S_{0}\left(X_{1}, X_{2}\right), B \in \mathcal{L}\left(X_{2}, X_{3}\right)$ and $C \in \mathcal{L}\left(X_{3}, X_{1}\right)$ , then $B A \in S_{0}\left(X_{1}, X_{3}\right), A C \in S_{0}\left(X_{3}, X_{2}\right)$ ;

(4)

if $A \in S_{0}\left(X_{1}, X_{2}\right)$ , then $A^{\prime } \in S_{0}\left(X_{2}^{*}, X_{1}^{*}\right)$ , moreover $\operatorname {dim} R(A) = \operatorname {dim} R\left(A^{\prime }\right)$ .

Problem 10.35

(1)(2)

Let $X_{1}, X_{2}$ be normed linear spaces, $A: X_{1} \rightarrow X_{2}$ be a linear operator. Is it correct that $A \in S_{\infty }\left(X_{1}, X_{2}\right)$ , if

?

(1)

$\operatorname {dim} X_{1} < +\infty$ ;

(2)

$\operatorname {dim} X_{2} < +\infty$ ?

Problem 10.36

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

Which of the following operators in $l_{2}$ are compact:

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(1)

$A x = \left(0, x_{1}, x_{2}, \ldots , x_{n}, \ldots \right)$ ;

(2)

$A x = \left(0, x_{1}, \frac{x_{2}}{2}, \ldots , \frac{x_{n}}{n}, \ldots \right)$ ;

(3)

$A x = \left(x_{2}, x_{3}, \ldots , x_{100}, 0,0, \ldots \right)$ ;

(4)

$A x = \left(x_{3}, \frac{x_{4}}{2}, \ldots , \frac{x_{n}}{n-2}, \ldots \right)$ ?

Problem 10.37

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

Find the necessary and sufficient condition on a bounded sequence $\left\{ \alpha_{n}: n \geq 1\right\} \subset \mathbb {C}$ under which the operator $A x = \left(\alpha_{1} x_{1}, \alpha_{2} x_{2}, \ldots \right), x \in X$ , is compact in the space $X$ , if

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(1)

$X = l_{p}, 1 \leq p < +\infty$ ;

(2)

$X = l_{\infty }$ ;

(3)

$X = c$ ;

(4)

$X = c_{0}$ .

Problem 10.38

Find the necessary and sufficient condition for the compactness of the operator $A: l_{2} \rightarrow l_{2}$ , which is defined by a Jacobi matrix

\mathcal{A} =\left[\begin{smallmatrix} \alpha _{1} & \beta _{1} & 0 & 0 & 0 & \ldots \\ \gamma _{1} & \alpha _{2} & \beta _{2} & 0 & 0 & \ldots \\ 0 & \gamma _{2} & \alpha _{3} & \beta _{3} & 0 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \end{smallmatrix}\right], \text{ where }\left\{ \alpha _{j}, \beta _{j}, \gamma _{j}: j \geq 1\right\} \subset \mathbb {K}.

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Problem 10.39

(1)(2)

Is it necessary that $A \in S_{\infty }\left(X_{1}, X_{2}\right)$ , if

?

(1)

$X_{1}$ is an NLS, $X_{2}$ is a Banach space, $\left\{ A_{n}: n \geq 1\right\} \subset S_{\infty }\left(X_{1}, X_{2}\right)$ and $A_{n} \xrightarrow {s} A$ ;

(2)

$X_{1}, X_{2}$ are normed linear spaces, $\left\{ A_{n}: n \geq 1\right\} \subset S_{\infty }\left(X_{1}, X_{2}\right)$ and $A_{n} \rightrightarrows A$ ?

Problem 10.40

Let $K \in C\left(\left[a, b\right]^{2}\right)$ and

(A x)(t) = \int _{a}^{t} K(t, s) x(s) d s, t \in [a, b], x \in C(\left[a, b\right])

Prove that $A \in S_{\infty }(C(\left[a, b\right]))$ .

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Problem 10.41

Which of the following operators $A: C(\left[0,1\right]) \rightarrow C(\left[0,1\right])$ are compact:

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(1)

$(A x)(t) = x(0)+t^{2} x(1)$ ;

(2)

$(A x)(t) = \int_{0}^{t} s^{3} x(s) d s$ ;

(3)

$(A x)(t) = \int_{0}^{1} e^{i t s} x(s) d s$ ;

(4)

$(A x)(t) = t x(t)$ ;

(5)

$(A x)(t) = x\left(t^{3}\right) ;$

(6)

$(A x)(t) = 3 x(t)+\int_{0}^{1} e^{t s} x(s) d s$ ;

(7)

$(A x)(t) = \frac{1}{2}(x(t)+x(1-t))$ ;

(8)

$(A x)(t) = \int_{0} x\left(s^{2}\right) d s$ ?

Problem 10.42

(1)(2)

Prove that operator $A: X_{1} \rightarrow X_{2}$ , which is defined by the formula

(A x)(t) = \int _{a}^{b} K(t, s) x(s) d s, t \in [a, b], x \in X_{1}

is compact, if

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(1)

$X_{1} = L_{2}(\left[a, b\right]), X_{2} = C(\left[a, b\right]), K \in C\left(\left[a, b\right]^{2}\right) ;$

(2)

$X_{1} = X_{2} = L_{2}(\left[a, b\right]), K \in L_{2}\left(\left[a, b\right]^{2}\right)$ .

Problem 10.43

Which of the operators $A: L_{2}(\left[0,1\right]) \rightarrow L_{2}(\left[0,1\right])$ are compact:

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(1)

$(A x)(t) = \int_{0}^{1} t s x(s) d s$ ;

(2)

$(A x)(t) = p(t) \int_{0}^{1} q(s) x(s) d s$ , where $p, q \in L_{2}(\left[0,1\right])$ ;

(3)

$(A x)(t) = x(t)+\int_{0}^{1}\left(t s^{2}+s\right) x(s) d s$ ;

(4)

$(A x)(t) = \int_{0}^{t} x(s) d s$ ;

(5)

$(A x)(t) = \int_{0}^{t} t s x(s) d s$ ?

Problem 10.44

(1)(2)

Under what conditions on a function $a$ is the operator $(A x)(t) = a(t) x(t)$ compact

?

(1)

in the space $C(\left[0,1\right])$ , if $a \in C(\left[0,1\right])$ ?

(2)

in the space $L_{2}(\left[0,1\right])$ , if $a \in L_{\infty }(\left[0,1\right])$ ?

Problem 10.45

(1)(2)(3)

Prove that the operator of embedding an NLS $X_{1}$ into an NLS $X_{2}$ , i.e., the operator $A: X_{1} \rightarrow X_{2}$ , defined as $A x = x, x \in X_{1}$ , is compact if

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(1)

$X_{1} = C^{1}(\left[a, b\right]), X_{2} = C(\left[a, b\right])$ ;

(2)

$X_{1} = C^{k+1}(\left[a, b\right]), X_{2} = C^{k}(\left[a, b\right]), k \in \mathbb {N}$ ;

(3)

$X_{1} = C^{1}(\left[a, b\right]), X_{2} = L_{p}(\left[a, b\right]), 1 \leq p \leq +\infty$ .

Problem 10.46

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

Is the operator of embedding an NLS $X_{1}$ into an NLS $X_{2}$ compact if

?

(1)

$X_{1} = l_{1}, X_{2} = l_{2}$ ;

(2)

$X_{1} = l_{1}, X_{2} = c_{0}$ ;

(3)

$X_{1} = C(\left[0,1\right]), X_{2} = L_{2}(\left[0,1\right])$ ;

(4)

$X_{1} = L_{2}(\left[0,1\right]), X_{2} = L_{1}(\left[0,1\right])$ ?

Problem 10.47

Under what conditions on the sequence $\alpha = \left\{ \alpha_{n}: n \geq 1\right\} \subset [1,+\infty )$ is the operator of embedding the space $l_{2, \alpha }$ (see problem $2.6^{\circ }$ ) into $c_{0}$ compact?

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Problem 10.48

(1)(2)(3)

Is the operator $A: X_{1} \rightarrow X_{2},(A x)(t) = x^{\prime }(t), t \in [0,1]$ , compact if

?

(1)

$X_{1} = C^{1}(\left[0,1\right]), X_{2} = C(\left[0,1\right])$ ;

(2)

$X_{1} = C^{2}(\left[0,1\right]), X_{2} = C^{1}(\left[0,1\right])$ ;

(3)

$X_{1} = C^{2}(\left[0,1\right]), X_{2} = C(\left[0,1\right])$ ?

Problem 10.49

Prove that the operator of orthogonal projection in a Hilbert space is compact if and only if this operator is finite-dimensional.

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Problem 10.50

(1)(2)

Provide an example of an operator $A \in \mathcal{L}\left(l_{2}\right)$ such that

?

(1)

$A \notin S_{\infty }\left(l_{2}\right)$ , but $A^{2} \in S_{\infty }\left(l_{2}\right)$ ;

(2)

$A^{k} \notin S_{\infty }\left(l_{2}\right), 1 \leq k \leq m-1$ , but $A^{m} \in S_{\infty }\left(l_{2}\right)$ , where $m \geq 3$ is fixed.

Problem 10.51

(1)(2)(3)

Let $H$ be a Hilbert space, $A, B \in \mathcal{L}(H)$ . Is it true that

?

(1)

if $A+B \in S_{\infty }(H)$ , then $A, B \in S_{\infty }$ ;

(2)

if $A+B, A-B \in S_{\infty }(H)$ , then $A, B \in S_{\infty }$ ;

(3)

if $A+B \in S_{\infty }(H), R(A) \perp R(B)$ , then $A, B \in S_{\infty }$ ?

Problem 10.52

(1)(2)

Let $H$ be a Hilbert space, $A: H \rightarrow H$ be a linear operator. Is it necessary that $A \in \mathcal{L}(H)$ , if

?

(1)

$A B \in \mathcal{L}(H)$ for all $B \in S_{\infty }(H)$ ;

(2)

$B A \in \mathcal{L}(H)$ for all $B \in S_{\infty }(H)$ ?

Problem 10.53

Prove that the range of every compact operator is a separable set.

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Problem 10.54

Let $c_{0}, c_{1}, \ldots , c_{n} \in \mathbb {C}$ . Can a compact operator $A$ in an infinite-dimensional NLS satisfy the equation $\sum_{k = 0}^{n} c_{k} A^{k} = 0$ ? (Here $A^{0} = I$ .)

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Problem 10.55

(1)(2)

Can a compact operator $A$ in an infinite-dimensional Banach space $X$ have an algebraic inverse, which is defined

?

(1)

on $X$ ;

(2)

on $R(A)$ ?

Problem 10.56

Let $X$ be an NLS, $A \in \mathcal{L}(X)$ and there exists $c > 0$ such that $\left\| A x\right\| \geq c\left\| x\right\|$ for all $x \in X$ . Under what conditions on $X$ can the operator $A$ be compact?

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Problem 10.57

Let $X_{1}, X_{2}$ be normed linear spaces, $\operatorname {dim} X_{1} = +\infty$ and $A \in S_{\infty }\left(X_{1}, X_{2}\right)$ . Prove that there exists a sequence $\left\{ x_{n}: n \geq 1\right\} \subset X_{1}$ such that $\left\| x_{n}\right\| = 1$ and $A x_{n} \rightarrow 0, n \rightarrow \infty$ .

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Problem 10.58

(1)(2)

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(1)

A set $M$ is called nowhere dense in a metric space if its closure has no interior points. Prove that in an infinite-dimensional NLS $X$ every precompact set is nowhere dense.

(2)

A subset in a metric space is called a set of the first category, if it is a countable union of nowhere dense sets. Show that if $X_{1}, X_{2}$ are normed linear spaces, $\operatorname {dim} X_{2} = +\infty$ and $A \in S_{\infty }\left(X_{1}, X_{2}\right)$ , then $R(A)$ is a set of the first category in $X_{2}$ .

Problem 10.59

(1)(2)

Show that $C^{1}(\left[a, b\right])$ is a set of the first category in a Banach space $X$ , if

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(1)

$X = C(\left[a, b\right])$ ;

(2)

$X = L_{p}(\left[a, b\right]), 1 \leq p \leq +\infty$ .

Problem 10.60

Let $X_{1}$ be an NLS, $X_{2}$ be a Banach space, and $A \in S_{\infty }\left(X_{1}, X_{2}\right)$ be such that the set $R(A)$ is closed in $X_{2}$ . Show that $\operatorname {dim} R(A) < +\infty$ .

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Problem 10.61

Show that in a Hilbert space the subspace that lies in the range of a compact operator is finite-dimensional.

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Problem 10.62

Let $M$ be a subspace of $C(\left[a, b\right])$ such that $M \subset C^{1}(\left[a, b\right])$ . Show that $M$ is finite-dimensional.

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Problem 10.63

Let $H$ be a Hilbert space and $A \in \mathcal{L}(H)$ . Prove that the operator $A$ is compact if and only if there exists a sequence of finite-dimensional linear operators which converges uniformly to $A$ .

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Problem 10.64

Let $H$ be a Hilbert space, $A \in \mathcal{L}(H)$ . Prove that operators $A, A^{*}, A A^{*}$ , and $A^{*} A$ are simultaneously compact or not compact.

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Problem 10.65

(1)(2)(3)

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

?

(1)

if $A^{2} \in S_{\infty }(H)$ and $A$ is normal, then $A \in S_{\infty }(H)$ ;

(2)

if $A^{n} \in S_{\infty }(H)$ for some $n > 1$ and $A$ is normal, then $A \in S_{\infty }(H)$ ;

(3)

if $A \in S_{\infty }(H)$ and $A \geq 0$ , then $\sqrt{A} \in S_{\infty }(H)$ (see the definition of the operator $\sqrt{A}$ in problem 9.52*).

Remark.

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For a nonnegative operator $A \in \mathcal{L}(H)$ , the square root $\sqrt{A}$ is the unique nonnegative operator $B \in \mathcal{L}(H)$ such that $B^{2} = A$ (existence and uniqueness are proved in problem 9.52*).

Problem 10.66

Let $X$ be a Banach space, and the operator $A \in \mathcal{L}(X)$ be such that

\forall \varepsilon > 0 \exists B_{\varepsilon } \in S_{0}(X) \exists C_{\varepsilon } \in \mathcal{L}(X),\left\| C_{\varepsilon }\right\| < \varepsilon : A = B_{\varepsilon }+C_{\varepsilon }.

Prove that $A \in S_{\infty }(X)$ .

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Problem 10.67

Let $\left\{ e_{n}: n \geq 1\right\}$ be an orthonormal basis in a Hilbert space $H, A \in \mathcal{L}(H)$ and

\sup \left\{ \left\| A x\right\| \mid x \perp e_{k}, 1 \leq k \leq n ;\left\| x\right\| = 1\right\} \rightarrow 0, n \rightarrow \infty .

Prove that $A$ is a compact operator.

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Problem 10.68

Let $X_{1}$ be a reflexive Banach space (for example, a Hilbert space), $X_{2}$ be an NLS, and $A \in S_{\infty }\left(X_{1}, X_{2}\right)$ . Prove that the set $A(\bar{B}(0,1))$ is compact in $X_{2}$ .

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Problem 10.69

(1)(2)

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(1)

Let $H$ be a Hilbert space, $A \in S_{\infty }(H)$ . Prove that there exists $h \in H \backslash \left\{ 0\right\}$ such that $\left\| A h\right\| = \left\| A\right\| \cdot \left\| h\right\|$ .

(2)

Provide an example of a Banach space $X$ and an operator $A \in S_{\infty }(X)$ for which the statement of item (1) is false.

Problem 10.70

(1)(2)

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(1)

Let $X_{1}$ be a reflexive Banach space, $X_{2}$ be an NLS. Prove that $A \in \mathcal{L}\left(X_{1}, X_{2}\right)$ is compact if and only if it transforms every weakly convergent sequence into a strongly convergent one.

(2)

Provide an example of Banach spaces $X_{1}, X_{2}$ and an operator $A \in \mathcal{L}\left(X_{1}, X_{2}\right)$ , which transforms every weakly convergent sequence into a strongly convergent one, but is not compact.

Problem 10.71

Let $1 < p < +\infty$ and $A \in \mathcal{L}\left(L_{p}(\left[a, b\right]), C(\left[a, b\right])\right)$ . Prove that $A \in S_{\infty }\left(L_{p}(\left[a, b\right]), L_{p}(\left[a, b\right])\right)$ .

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Problem 10.72

Let $H$ be a Hilbert space, $A \in \mathcal{L}(H)$ and $A \geq 0$ . Prove that the operator $A$ is compact if and only if for every sequence $\left\{ x_{n}: n \geq 1\right\} \subset H$ which converges weakly to zero the relation $\left(A x_{n}, x_{n}\right) \rightarrow 0, n \rightarrow \infty$ , holds.

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Problem 10.73

(1)(2)

Let $H$ be a Hilbert space, $A \in \mathcal{L}(H), B \in S_{\infty }(H)$ . Prove that $A \in S_{\infty }(H)$ in each of the following cases:

?

(1)

$0 \leq A \leq B$ ;

(2)

$A^{*} A \leq B^{*} B$ .

Problem 10.74

(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 $A^{\prime } \in S_{\infty }\left(X_{2}^{*}, X_{1}^{*}\right)$ .

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Problem 10.75

(1)(2)

Let $X$ be a reflexive Banach space. Prove that

?

(1)

$S_{\infty }\left(X, l_{1}\right) = \mathcal{L}\left(X, l_{1}\right)$ ;

(2)

$S_{\infty }\left(c_{0}, X\right) = \mathcal{L}\left(c_{0}, X\right)$ 。

Problem 10.76

(1)(2)

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(1)

Let $X, Y$ be Banach spaces, $\left\{ A, A_{n}: n \geq 1\right\} \subset \mathcal{L}(X, Y), A_{n} \xrightarrow {s} A, n \rightarrow \infty$ , and $M$ be a precompact set in the space $X$ . Prove that $\left\{ A_{n} x: n \geq 1\right\}$ converges to $A x$ uniformly on $M$ .

(2)

Let $X$ be a Banach space, $\left\{ f, f_{n}: n \geq 1\right\} \subset X^{*}, f_{n} \xrightarrow {w-*} f, n \rightarrow \infty$ , and $M$ be a precompact set in the space $X$ . Prove that $\left\{ f_{n}(x): n \geq 1\right\}$ converges to $f(x)$ uniformly on $M$ .

Problem 10.77

(1)(2)

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(1)

Let $X$ be a separable Banach space, $\left\{ x_{n}: n \geq 1\right\}$ be a countable sequence, which is dense in $X, B$ be the ball $\bar{B}(0,1)$ in $X^{*}$ . Put

\rho (f, g) = \sum _{n = 1}^{\infty } \frac{1}{2^{n}} \cdot \frac{\left|f\left(x_{n}\right)-g\left(x_{n}\right)\right|}{1+\left|f\left(x_{n}\right)-g\left(x_{n}\right)\right|}, f, g \in B.

Prove that $(B, \rho )$ is a compact metric space, and the convergence in $(B, \rho )$ is equivalent to weak-* convergence.

(2)

(Gelfand). Prove that for a set $M$ in a Banach space $X$ to be compact it is necessary, and in the case of a separable space $X$ it is also sufficient, that for each sequence $\left\{ f_{n}: n \geq 1\right\} \subset X^{*}$ which converges weakly-* to zero, the convergence $f_{n}(x) \rightarrow 0, n \rightarrow \infty$ , is uniform for $x \in M$ .

Problem 10.78

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 $X_{1}^{*}$ .

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Problem 10.79

Let $X$ be a Banach space, $\left\{ A_{n}: n \geq 1\right\} \subset S_{\infty }(X)$ , and $A_{n} \xrightarrow {s} I, n \rightarrow \infty$ . Prove that a set $M \subset X$ is precompact if and only if the following conditions are fulfilled:

(a) $M$ is a bounded set;

(b) $A_{n} x \rightarrow x, n \rightarrow \infty$ , uniformly in $x \in M$ .

Verify that condition (b) can be replaced by condition (b') $\forall \varepsilon > 0 \exists n \in \mathbb {N} \forall x \in M:\left\| A_{n} x-x\right\| < \varepsilon$ , and condition $\left\{ A_{n}: n \geq 1\right\} \subset S_{\infty }(X)$ can be replaced by condition

\exists k \in \mathbb {N}:\left\{ A_{n}^{k}: n \geq 1\right\} \subset S_{\infty }(X).

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Problem 10.80

Let $1 \leq p < +\infty , x \in L_{p}(\left[a, b\right]), h > 0$ . The function

\left(A_{h} x\right)(t) = x_{h}(t) = \frac{1}{2 h} \int _{t-h}^{t+h} x(s) d s, t \in [a, b]

where $x(t) = 0, t \in \mathbb {R} \backslash [a, b]$ , is called the Steklov mean function. Prove that

?

(1)

$x_{h} \in C(\left[a, b\right])$ and $\left|x_{h}(t)\right| \leq \frac{\left\| x\right\|_{p}}{(2 h)^{\frac{1}{p}}}, t \in [a, b]$ ;

(2)

$\left\| x_{h}\right\|_{p} \leq \left\| x\right\|_{p}$ for $p = 1$ ;

(3)

$\left\| x_{h}\right\|_{p} \leq \left\| x\right\|_{p}$ for $1 < p < +\infty$ ;

(4)

$\left\| A_{h} x-x\right\|_{p} \rightarrow 0, h \rightarrow 0+;$

(5)

$A_{h} \in S_{\infty }\left(L_{p}(\left[a, b\right]), C(\left[a, b\right])\right), 1 < p < +\infty$ ;

(6)

$A_{h}^{2} \in S_{\infty }\left(L_{1}(\left[a, b\right]), C(\left[a, b\right])\right)$ ;

(7)

$A_{h} \in S_{\infty }\left(L_{p}(\left[a, b\right])\right), 1 < p < +\infty$ , and $A_{h}^{2} \in S_{\infty }\left(L_{1}(\left[a, b\right])\right)$ .

Problem 10.81

Prove that a set $M$ is precompact in the space $L_{p}(\left[a, b\right]), 1 \leq p < +\infty$ , if and only if the following conditions are fulfilled:

(a) $M$ is a bounded set;

(b) $\forall \varepsilon > 0 \exists h > 0 \forall x \in M:\left\| x_{h}-x\right\|_{p} < \varepsilon$ .

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Remark.

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Condition (a) follows from condition (b), see problem 12.29.

Problem 10.82

Prove that a set $M$ is precompact in the space $L_{p}(\left[a, b\right]), 1 \leq p < +\infty$ , if and only if the following conditions are fulfilled:

(a) $M$ is a bounded set;

(b) the set $M$ is equicontinuous in $p$ -mean i.e.,

\forall \varepsilon > 0 \exists \delta > 0 \forall h \in (0, \delta ) \forall x \in M: \int _{a}^{b}\left|x(t+h)-x(t)\right|^{p} d t < \varepsilon ^{p}

(here $x(t+h) = 0$ , if $t+h > b$ ).

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Remark.

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Condition (a) follows from condition (b), see problem 12.29.

Problem 10.83

Let $B_{n}, n \geq 1$ , be Bernstein operators:

\left(B_{n} x\right)(t) = \sum _{k = 0}^{n}\binom {n}{k} x\left(\frac{k}{n}\right) t^{k}(1-t)^{n-k}, t \in [0,1], x \in C(\left[0,1\right]).

Is it true that $B_{n} \rightrightarrows I$ ?

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Problem 10.84

Let $X, Y$ and $Z$ be Banach spaces, $A \in S_{\infty }(X, Z), B \in \mathcal{L}(Y, Z)$ , and $R(B) \subset R(A)$ . Prove that $B \in S_{\infty }(Y, Z)$ .

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Problem 10.85

Let $X_{1}, X_{2}$ be normed linear spaces, $M \subset X_{1}$ be a linear set which is dense in $X_{1}$ , and $A \in \mathcal{L}\left(X_{1}, X_{2}\right)$ . Prove that if the restriction of the operator $A$ on $M$ belongs to $S_{\infty }\left(M, X_{2}\right)$ , then $A \in S_{\infty }\left(X_{1}, X_{2}\right)$ .

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Problem 10.86

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

Let $W_{p}^{1}(\left[a, b\right]), p \geq 1$ , be the completion of the space $C^{1}(\left[a, b\right])$ in the norm $\left\| x\right\|_{W_{p}^{1}} = \left(\int_{a}^{b}\left|x(t)\right|^{p} d t+\int_{a}^{b}\left|x^{\prime }(t)\right|^{p} d t\right)^{\frac{1}{p}}, x \in C^{1}(\left[a, b\right])$ . Prove that

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(1)

for every function $x \in C^{1}(\left[a, b\right])$ there exists $\theta \in [a, b]$ such that

x(t) = \int _{\theta }^{t} x^{\prime }(s) d s+\frac{1}{b-a} \int _{a}^{b} x(s) d s, t \in [a, b]

(2)

for $p \geq 1$ the inequality $\left\| x\right\|_{C} \leq c\left\| x\right\|_{W_{p}^{1}}, x \in C^{1}(\left[a, b\right])$ , holds, where $c > 0$ is a constant, and $\left\| \cdot \right\|_{C}$ and $\left\| \cdot \right\|_{W_{p}^{1}}$ are norms in $C(\left[a, b\right])$ and $W_{p}^{1}(\left[a, b\right])$ , respectively;

(3)

for $p \geq 1$ we have $W_{p}^{1}(\left[a, b\right]) \subset C(\left[a, b\right])$ and the embedding operator $A$ : $W_{p}^{1}(\left[a, b\right]) \rightarrow C(\left[a, b\right]), A x = x$ , is continuous;

(4)

for $p > 1$ the embedding operator $A: W_{p}^{1}(\left[a, b\right]) \rightarrow C(\left[a, b\right]), A x = x$ , is compact.

Problem 10.87

(1)(2)

Let $p \geq 1$ . Prove that

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(1)

there exists a unique continuous linear operator $D: W_{p}^{1}(\left[a, b\right]) \rightarrow L_{p}(\left[a, b\right])$ such that $D x(t) = x^{\prime }(t), t \in [a, b]$ , for all $x \in C^{1}(\left[a, b\right])$ ;

(2)

the operator $D$ is not compact.

Problem 10.88

Let $(T, \mathcal{F}, \mu )$ be a space with a finite measure, $p > 1, \frac{1}{p}+\frac{1}{q} = 1, K: T \times T \rightarrow \mathbb {C}$ be a measurable function such that $\operatorname {ess}_{t \in T} \int_{T}\left|K(t, s)\right|^{q} d \mu (s) < +\infty$ . Prove that the operator $A: L_{p}(T, \mu ) \rightarrow L_{p}(T, \mu )$ , defined by the formula $(A x)(t) = \int_{T} K(t, s) x(s) d \mu (s), t \in T$ , is compact.

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Problem 10.89

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

Let $(T, \mathcal{F}, \mu )$ be a space with a measure, $K: T \times T \rightarrow \mathbb {R}$ be a measurable function, $c_{1} = \operatorname {ess} \sup_{s \in T} \int_{T}\left|K(t, s)\right| d \mu (t), c_{2} = \operatorname {ess} \sup_{t \in T} \int_{T}\left|K(t, s)\right| d \mu (s)$ .

Prove that the operator $(A x)(t) = \int_{T} K(t, s) x(s) d \mu (s), t \in T$ , has the following properties:

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(1)

if $c_{1} < +\infty$ , then $A \in \mathcal{L}\left(L_{1}(T, \mu )\right)$ and $\left\| A\right\| \leq c_{1}$ ;

(2)

if $c_{1} < +\infty , c_{2} < +\infty$ , then $A \in \mathcal{L}\left(L_{p}(T, \mu )\right)$ and $\left\| A\right\| \leq c_{1}^{\frac{1}{p}} c_{2}^{\frac{1}{q}}, 1 < p < +\infty ;$

(3)

under the conditions of item (2) the operator $A$ is not necessarily compact in the space $L_{p}(T, \mu ), 1 \leq p < +\infty$ , even if $\mu (T) < +\infty$ ;

(4)

if $1 < p < +\infty , c_{2} < +\infty , \mu (T) < +\infty$ and

c_{1, n} = \underset {s \in T}{\operatorname {ess} \sup } \int _{B_{n}(s)}\left|K(t, s)\right| d \mu (t) \rightarrow 0, n \rightarrow \infty

where $B_{n}(s) = \left\{ t \in T \mid \left|K(t, s)\right| > n\right\} , n \geq 1, s \in T$ , then $A \in S_{\infty }\left(L_{p}(T, \mu )\right)$ .

Problem 10.90

Let $K: \mathbb {R} \rightarrow \mathbb {C}$ be a $2 \pi$ -periodic Lebesgue measurable function and $K \in L_{1}(\left[0,2 \pi \right])$ . Prove that for $1 < p < +\infty$ the operator

(A x)(t) = \int _{0}^{2 \pi } K(t-s) x(s) d s, t \in [0,2 \pi ], x \in L_{p}(\left[0,2 \pi \right])

is compact.

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Problem 10.91

(1)(2)(3)

Prove the compactness of the operator $A: X \rightarrow X$ , defined by the formula $(A x)(t) = \int_{a}^{b} K(t, s) x(s) d s, t \in [a, b], x \in X$ , if

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(1)

$X = C(\left[a, b\right]), K:\left[a, b\right]^{2} \rightarrow \mathbb {R}$ is a Lebesgue measurable function such that $K(t, \cdot ) \in L_{1}(\left[a, b\right])$ for all $t \in [a, b]$ , and

\forall t_{0} \in [a, b]: \int _{a}^{b}\left|K(t, s)-K\left(t_{0}, s\right)\right| d s \rightarrow 0, t \rightarrow t_{0}

(2)

$X = C(\left[a, b\right]), K(t, s) = \frac{K_{0}(t, s)}{\left|t-s\right|^{\alpha }},(t, s) \in [a, b]^{2}, t \neq s$ , where $K_{0} \in C\left(\left[a, b\right]^{2}\right), \alpha < 1 ;$

(3)

$X = L_{2}(\left[a, b\right]), K(t, s) = \frac{K_{0}(t, s)}{\left|t-s\right|^{\alpha }},(t, s) \in [a, b]^{2}, t \neq s$ , where $K_{0} \in L_{\infty }\left(\left[a, b\right]^{2}\right), \alpha < 1$ .

Problem 10.92

Let $u \in L_{1}(\mathbb {R})$ . Prove that the operator

(A x)(t) = \int _{\mathbb {R}} u(t-s) x(s) d s, t \in \mathbb {R}, x \in L_{2}(\mathbb {R})

is linear and continuous in $L_{2}(\mathbb {R})$ . Under what conditions on the function $u$ is the operator $A$ compact?

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Problem 10.93

Let $u \in L_{1}([0,+\infty ))$ and $(A x)(t) = \int_{0}^{t} u(t-s) x(s) d s, t \in [0,+\infty ), x \in L_{2}([0,+\infty ))$ . Prove that $A \in \mathcal{L}\left(L_{2}([0,+\infty ))\right)$ . Under what conditions on the function $u$ is the operator $A$ compact?

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Problem 10.94

(1)(2)

Let $u \in L_{1}([0,+\infty ))$ and $(A x)(t) = \int_{0}^{+\infty } u(t+s) x(s) d s, t \in [0,+\infty ), x \in L_{2}([0,+\infty ))$ . Prove that

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(1)

$A \in \mathcal{L}\left(L_{2}([0,+\infty ))\right)$ ;

(2)

if $\int_{0}^{+\infty } t\left|u(t)\right|^{2} d t < +\infty$ , then $A \in S_{\infty }\left(L_{2}([0,+\infty ))\right)$ .

Problem 10.95

(1)(2)

Let $(A x)(t) = \int_{0}^{+\infty } \frac{x(s)}{t+s} d s, t \in [0,+\infty ), x \in L_{2}([0,+\infty ))$ . Prove that

?

(1)

$A \in \mathcal{L}\left(L_{2}([0,+\infty ))\right)$ ;

(2)

$A \notin S_{\infty }\left(L_{2}([0,+\infty ))\right)$ .

Problem 10.96

(1)(2)

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(1)

Prove that the operator $A: L_{1}(\left[0,1\right]) \rightarrow L_{1}(\left[0,1\right])$ , defined by the formula $(A x)(t) = \int_{0}^{t} x(s) d s, t \in [0,1], x \in L_{1}(\left[0,1\right])$ , is compact.

(2)

Provide an example of a bounded Borel function $K:\left[0,1\right]^{2} \rightarrow \mathbb {R}$ such that the operator $A: L_{1}(\left[0,1\right]) \rightarrow L_{1}(\left[0,1\right])$ , defined by the formula

(A x)(t) = \int _{0}^{1} K(t, s) x(s) d s, t \in [0,1], x \in L_{1}(\left[0,1\right])

is not compact.

Problem 10.97

(1)(2)

Let $C_{b}(\mathbb {R})$ be the linear space of continuous functions which are bounded on $\mathbb {R}$ endowed with the norm $\left\| x\right\| = \sup_{t \in \mathbb {R}}\left|x(t)\right|, x \in C_{b}(\mathbb {R})$ , and let a function $K \in C\left(\mathbb {R}^{2}\right)$ be such that $\sup_{t \in \mathbb {R} \mathbb {R}}\left|K(t, s)\right| d s < \infty$ . Consider the integral operator

(A x)(t) = \int _{\mathbb {R}} K(t, s) x(s) d s, t \in \mathbb {R}, x \in C_{b}(\mathbb {R})

Prove that

?

(1)

$A \in \mathcal{L}\left(C_{b}(\mathbb {R})\right)$ ;

(2)

it is not necessarily that $A \in S_{\infty }\left(C_{b}(\mathbb {R})\right)$ .

Problem 10.98

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

Let $X = C(\left[0,1\right])$ or $X = L_{p}([0,+\infty )), 1 < p < +\infty$ . The operator $A$ is defined by the formula $(A x)(t) = \frac{1}{t} \int_{0}^{t} x(s) d s, t > 0, x \in X$ (in the case of $X = C(\left[0,1\right])$ additionally $(A x)(0): = x(0))$ . Prove that

?

(1)

$A \in \mathcal{L}(C(\left[0,1\right])) ;$

(2)

$A \notin S_{\infty }(C(\left[0,1\right]))$ ;

(3)

$A \in \mathcal{L}\left(L_{p}([0,+\infty ))\right) ;$

(4)

$A \notin S_{\infty }\left(L_{p}([0,+\infty ))\right)$ .