Math Problems Vault

Problem 7.4

Prove the following statements:

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(1)

if the limit exists, then it is unique for any operator convergence;

(2)

uniform convergence implies strong convergence;

(3)

strong convergence implies weak convergence;

(4)

the converse statements to the statements (2) and (3) are not true;

(5)

in a finite-dimensional space uniform, strong and weak convergences are equivalent.

Problem 7.5

What does the uniform, strong and weak operator convergence look like for operators from $\mathcal{L}(X, \mathbb {K})$ , where $X$ is an NLS, i.e., for functionals?

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Problem 7.6

Let $X_{1}$ and $X_{2}$ be normed linear spaces, $\left\{ A_{n}: n \geq 1\right\} \subset \mathcal{L}\left(X_{1}, X_{2}\right), A \in \mathcal{L}\left(X_{1}, X_{2}\right)$ . Prove that for the convergence $A_{n} \xrightarrow {s} A, n \rightarrow \infty$ , it is sufficient, and if $X_{1}$ is a Banach space, then it is also necessary, that the following conditions are fulfilled:

(a) the sequence $\left\{ \left\| A_{n}\right\| : n \geq 1\right\}$ is bounded;

(b) there exists a total set $M \subset X_{1}$ such that $A_{n} x \rightarrow A x, n \rightarrow \infty$ , in $X_{2}$ for all $x \in M$ .

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Problem 7.7

Formulate and prove theorems, analogous to problem $7.6^{\circ }$ , for weak operator convergence.

Problem 7.6: Let $X_{1}$ and $X_{2}$ be normed linear spaces, $\left\{ A_{n}: n \geq 1\right\} \subset \mathcal{L}\left(X_{1}, X_{2}\right), A \in \mathcal{L}\left(X_{1}, X_{2}\right)$ . Prove that for the convergence $A_{n} \xrightarrow {s} A, n \rightarrow \infty$ , it is sufficient, and if $X_{1}$ is a Banach space, then it is also necessary, that the following conditions are fulfilled:

(a) the sequence $\left\{ \left\| A_{n}\right\| : n \geq 1\right\}$ is bounded;

(b) there exists a total set $M \subset X_{1}$ such that $A_{n} x \rightarrow A x, n \rightarrow \infty$ , in $X_{2}$ for all $x \in M$ .

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Problem 7.8

Examine the sequence of operators $A_{n}: l_{p} \rightarrow l_{p}, n \geq 1$ , for uniform, strong and weak convergence.

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(1)

$A_{n} x = (\underbrace{0, \ldots , 0}_{n}, x_{1}, x_{2}, \ldots ), 1 \leq p < +\infty$ ;

(2)

$A_{n} x = \left(x_{n+1}, x_{n+2}, \ldots \right), 1 \leq p \leq +\infty ;$

(3)

$A_{n} x = (\underbrace{0, \ldots , 0}_{n-1}, x_{n}, 0,0, \ldots ), 1 \leq p < +\infty$ ;

(4)

$A_{n} x = (\underbrace{0, \ldots , 0}_{n-1}, x_{n}, x_{n+1}, \ldots ), 1 \leq p < +\infty$ ;

(5)

$A_{n} x = \left(x_{n}, x_{n-1}, x_{2}, \ldots x_{1}, 0,0, \ldots \right), 1 \leq p \leq +\infty ;$

(6)

$A_{n} x = \left(\frac{x_{1}}{n}, \frac{x_{2}}{n-1}, \ldots , \frac{x_{n}}{1}, 0,0, \ldots \right), 1 \leq p < +\infty ;$

(7)

$A_{n} x = (\underbrace{0, \ldots , 0}_{n-1}, \sqrt{n} x_{2 n}, 0,0, \ldots ), 1 \leq p \leq +\infty ;$

(8)

$A_{n} x = \left(\sum_{m = 1}^{n} x_{m}, 0,0, \ldots \right), 1 \leq p \leq +\infty ;$

(9)

$A_{n} x = \left(\frac{1+n}{1+n} x_{1}, \frac{1+n}{1+2 n} x_{2}, \frac{1+n}{1+3 n} x_{3}, \ldots , \frac{1+n}{1+n^{2}} x_{n}, 0,0, \ldots \right), 1 \leq p \leq +\infty ;$

(10)

$A_{n} x = \left(\sqrt[n]{n} x_{1}, \sqrt[n]{n} x_{2}, \ldots , \sqrt[n]{n} x_{n}, \ldots \right), 1 \leq p \leq +\infty ;$

(11)

$A_{n} x = \left(\sqrt[n]{n} x_{1}, \sqrt[n]{n} x_{2}, \ldots , \sqrt[n]{n} x_{n}, 0,0, \ldots \right), 1 \leq p < +\infty ;$

(12)

$A_{n} x = \left(\frac{x_{n}}{2 n}, \frac{x_{n+1}}{2 n+1}, \ldots , \frac{x_{2 n}}{3 n}, 0,0, \ldots \right), 1 \leq p \leq +\infty ;$

(13)

$A_{n} x = \left(\frac{1+n}{1+n} x_{1}, \frac{1+n}{1+2 n} x_{2}, \frac{1+n}{1+3 n} x_{3}, \ldots , \frac{1+n}{1+k n} x_{k}, \ldots \right), 1 \leq p \leq +\infty ;$

(14)

$A_{n} x = \left(\frac{1+n}{1+n} x_{1}, \frac{1+n}{1+2 n} x_{2}, \frac{1+n}{1+3 n} x_{3}, \ldots , \frac{1+n}{1+k n} x_{k}, 0,0, \ldots \right), 1 \leq p \leq +\infty$ , where $k \in \mathbb {N}$ is fixed.

Problem 7.9

Examine the sequence of operators from items (3), (6) and (11) of problem $7.8^{\circ }$ for uniform, strong or weak convergence in the case of $p = +\infty$ .

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Problem 7.10

Find the necessary and sufficient conditions for the sequence of complex numbers $\left\{ \alpha_{n}: n \geq 1\right\}$ , under which the sequence of operators $A_{n}: l_{p} \rightarrow l_{p}$ , $1 \leq p \leq +\infty , n \geq 1$ , converges uniformly, strongly, weakly, if

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(1)

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

(2)

$A_{n} x = (\underbrace{0,0, \ldots , 0}_{n-1}, \alpha_{n} x_{1}, 0,0, \ldots )$ ;

(3)

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

(4)

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

(5)

$A_{n} x = (\underbrace{0,0, \ldots , 0}_{n-1}, \alpha_{1} x_{1}, \alpha_{2} x_{2}, \ldots , \alpha_{n} x_{n}, 0,0, \ldots )$ ;

(6)

$A_{n} x = (\underbrace{0,0, \ldots , 0}_{n-1}, \alpha_{n} x_{1}, \alpha_{n-1} x_{2}, \ldots , \alpha_{1} x_{n}, 0,0, \ldots )$ ;

(7)

$A_{n} x = \left(\alpha_{n} x_{n}, \alpha_{n-1} x_{n+1}, \ldots , \alpha_{1} x_{2 n-1}, 0,0, \ldots \right)$ ;

(8)

$A_{n} x = \left(\alpha_{n} x_{1}, \alpha_{n+1} x_{2}, \ldots \right)$ ;

(9)

$A_{n} x = \left(\alpha_{n} x_{1}, \alpha_{n+1} x_{2}, \ldots , \alpha_{2 n-1} x_{n}, 0,0, \ldots \right)$ ;

(10)

$A_{n} x = \left(\alpha_{n} x_{1}, \alpha_{n-1} x_{2}, \ldots , \alpha_{1} x_{n}, 0,0, \ldots \right)$ .

Problem 7.11

Examine the sequence of operators $A_{n}: C(\left[0,1\right]) \rightarrow C(\left[0,1\right])$ for uniform, strong, and weak convergence.

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(1)

$\left(A_{n} x\right)(t) = t^{n}(1-t) x(t) ;$

(2)

$\left(A_{n} x\right)(t) = t^{n} x(t)$ ;

(3)

$\left(A_{n} x\right)(t) = e^{-n t} x(t)$ ;

(4)

$\left(A_{n} x\right)(t) = x(t) \cdot \sin n t$ ;

(5)

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

(6)

$\left(A_{n} x\right)(t) = x(t) \cdot \sin \frac{t}{n}$ ;

(7)

$\left(A_{n} x\right)(t) = \int_{0}^{1} \frac{(t+s)^{n}}{3} x(s) d s$ ;

(8)

$\left(A_{n} x\right)(t) = \int_{0}^{1} \frac{(t+s)^{n}}{3^{n}} x(s) d s$ ;

(9)

$\left(A_{n} x\right)(t) = \int_{0}^{t}\left(\sum_{k = 0}^{n} \frac{(-1)^{k} s^{2 k+1}}{(2 k+1)!}\right) x(s) d s ;$

(10)

$\left(A_{n} x\right)(t) = x\left(t^{1+\frac{1}{n}}\right)$ ;

(11)

$\left(A_{n} x\right)(t) = \left(t^{10 n}-t^{2 n}\right) x(t) ;$

(12)

$\left(A_{n} x\right)(t) = t \int_{0}^{1} x\left(\sin^{n} \frac{\pi }{2} s\right) d s$ ;

(13)

$\left(A_{n} x\right)(t) = t \int_{0}^{1} x(s) \sin^{n} \frac{\pi }{2} s d s$ ;

(14)

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

(15)

$\left(A_{n} x\right)(t) = \frac{\ln (1+n t)}{1+n t} x(t)$ ;

(16)

$\left(A_{n} x\right)(t) = \frac{\ln (1+t)}{1+n t} x(t)$ ;

(17)

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

(18)

$\left(A_{n} x\right)(t) = \int_{0}^{\frac{t^{n}}{2^{n}}} x(s) d s ;$

(19)

$\left(A_{n} x\right)(t) = \int_{0}^{t^{n}} x(s) d s ;$

(20)

$\left(A_{n} x\right)(t) = (1-t)^{n} \int_{0}^{t^{n}} x(s) d s$ ;

(21)

$\left(A_{n} x\right)(t) = n \int_{t}^{t+\frac{1}{n}} x(s) d s$ , where $x(s) = x(1)$ for $s > 1$ .

Problem 7.12

Examine the sequence of operators $A_{n}: L_{p}(T) \rightarrow L_{p}(T)$ for uniform, strong, and weak convergence.

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(1)

$T = \mathbb {R},\left(A_{n} x\right)(t) = x(t+n), 1 \leq p < +\infty ;$

(2)

$T = \mathbb {R},\left(A_{n} x\right)(t) = \begin{cases} x(t), & t \geq n, \\ 0, & t < n, \end{cases} \quad 1 \leq p < +\infty ;$

(3)

$T = \mathbb {R},\left(A_{n} x\right)(t) = \frac{1}{1+\left|t-n\right|} x(t), 1 \leq p < +\infty ;$

(4)

$T = \left[0,1\right],\left(A_{n} x\right)(t) = t^{n} \int_{0}^{1} s^{n} x(s) d s, 1 \leq p < +\infty ;$

(5)

$T = \left[0,1\right],\left(A_{n} x\right)(t) = x(t) \cos n t, p = 2$ ;

(6)

$T = \left[0,1\right],\left(A_{n} x\right)(t) = x(t) \cos e^{n} t, p = 2$ ;

(7)

$T = \left[0,1\right],\left(A_{n} x\right)(t) = x(t) \cos n e^{t}, p = 2$ ;

(8)

$T = \left[0,1\right],\left(A_{n} x\right)(t) = x(t) \cos e^{n t}, p = 2$ ;

(9)

$T = \mathbb {R},\left(A_{n} x\right)(t) = x(t) \cdot \chi_{\left[\ln n, 2 \ln n\right]}(t), 1 \leq p < +\infty ;$

(10)

$T = \left[0,1\right],\left(A_{n} x\right)(t) = \sqrt[n]{t} x(t), p = 1$ ;

(11)

$T = \left[0,1\right],\left(A_{n} x\right)(t) = \int_{0}^{1} t^{2}\left(s^{n}-s^{2 n}\right) x(s) d s, 1 \leq p < +\infty$ .

Problem 7.13

(1)(2)(3)

Let $\left\{ p_{n}: n \geq 1\right\} \subset C(\left[a, b\right])$ be a fixed sequence,

\left(A_{n} x\right)(t) = p_{n}(t) x(t), t \in [a, b], x \in C(\left[a, b\right]).

Under what conditions on the sequence $\left\{ p_{n}: n \geq 1\right\}$ the sequence of operators $\left\{ A_{n}: n \geq 1\right\}$ converges

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(1)

uniformly;

(2)

strongly;

(3)

weakly? Determine the form of the limit operator.

Problem 7.14

(1)(2)

Let $X_{1}, X_{2}$ be Banach spaces.

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(1)

Prove that the space $\mathcal{L}\left(X_{1}, X_{2}\right)$ is complete with respect to the strong convergence of operators, i.e., every sequence $\left\{ A_{n}: n \geq 1\right\} \subset \mathcal{L}\left(X_{1}, X_{2}\right)$ , for which $\left\{ A_{n} x: n \geq 1\right\}$ is a Cauchy sequence of elements from $X_{2}$ for each $x \in X_{1}$ , converges strongly to some operator $A \in \mathcal{L}\left(X_{1}, X_{2}\right)$ .

(2)

Under what conditions on the Banach spaces $X_{1}$ and $X_{2}$ is the space $\mathcal{L}\left(X_{1}, X_{2}\right)$ complete with respect to weak convergence?

Problem 7.15

In $L_{p}(\mathbb {R}), 1 \leq p < +\infty$ , define the shift operator

\left(A_{s} x\right)(t) = x(t+s), t \in \mathbb {R}, x \in L_{p}(\mathbb {R}), s \in \mathbb {R}

Let $s_{n} \rightarrow s, n \rightarrow \infty$ , and $s_{n} \neq s, n \geq 1$ . Prove that $A_{s_{n}} \xrightarrow {s} A_{s}$ , but $A_{s_{n}} \nRightarrow A_{s}$ .

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Problem 7.16

(1)(2)(3)

Let $X$ be the space $C^{1}(\left[0,1\right])$ with the norm $\left\| x\right\| = \max_{t \in [0,1]}\left|x(t)\right|, x \in C^{1}(\left[0,1\right])$ . Define the operator $A_{n}: X \rightarrow C(\left[0,1\right])$ by the formula $\left(A_{n} x\right)(t) = n\left(x\left(t+\frac{1}{n}\right)-x(t)\right), \quad t \in [0,1], \quad x \in X, \quad n \geq 1$ (if $t > 1$ , put $x(t) = x(1)+x_{-}^{\prime }(1)(t-1)$ .) Prove that

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(1)

for every $x \in X$ the sequence $\left\{ A_{n} x: n \geq 1\right\}$ converges in norm in $C(\left[0,1\right])$ ;

(2)

the sequence $\left\{ \left\| A_{n}\right\| : n \geq 1\right\}$ is not bounded;

(3)

the space $\mathcal{L}(X, C(\left[0,1\right]))$ is not complete with respect to the strong operator topology.

Do these statements contradict the Uniform Boundedness Principle?

Problem 7.17

Denote by $\widetilde{C}$ the space of all $2 \pi$ -periodic continuous functions on $\mathbb {R}$ . It can be identified with a subspace in $C(\left[-\pi , \pi \right])$ , which consists of functions $x$ such that $x(-\pi ) = x(\pi )$ . For any $n \geq 1$ and $t \in [-\pi , \pi ]$ put

\left(S_{n} x\right)(t) = \frac{1}{\pi } \int _{-\pi }^{\pi } D_{n}(t-s) x(s) d s

where $D_{n}(t) = \frac{1}{2}+\sum_{k = 1}^{n} \cos k t = \frac{\sin \frac{2 n+1}{2} t}{2 \sin \frac{t}{2}}$ is the Dirichlet kernel. Prove that

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(1)

$S_{n} \xrightarrow {s} I, S_{n} \nRightarrow I$ in $L_{2}(\left[-\pi , \pi \right])$ ;

(2)

$\int_{-\pi }^{\pi }\left|D_{n}(t)\right| d t > A \ln n, n \geq 1$ , for some constant $A > 0$ ;

(3)

there exist functions $x \in \widetilde{C}\left(x \in L_{1}(\left[-\pi , \pi \right])\right)$ , for which the Fourier series with respect to the trigonometric system does not converge to $x$ in $\widetilde{C}$ (in $L_{1}(\left[-\pi , \pi \right])$ , respectively);

(4)

for any $t_{0} \in [-\pi , \pi ]$ there exists a function $x \in \widetilde{C}$ , for which the Fourier series at the point $t_{0}$ does not converge to $x\left(t_{0}\right)$ ;

(5)

there exist functions $x \in C(\left[-\pi , \pi \right])$ and $y \in L_{1}(\left[-\pi , \pi \right])$ such that the series

\frac{a_{0} \alpha _{0}}{2}+\sum _{n = 1}^{\infty }\left(a_{n} \alpha _{n}+b_{n} \beta _{n}\right)

diverges, where $\left\{ a_{0}, a_{n}, b_{n}: n \geq 1\right\} ,\left\{ \alpha_{0}, \alpha_{n}, \beta_{n}: n \geq 1\right\}$ are the Fourier coefficients of the functions $x$ and $y$ , respectively.

Problem 7.18

Let $X_{1}$ be a Banach space, $X_{2}$ be an NLS and the mapping $B: X_{1} \times X_{1} \rightarrow X_{2}$ be linear and continuous in each variable. Prove that $B$ is joint continuous.

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Problem 7.19

Let $X$ be an NLS which consists of algebraic polynomials of one variable with a norm $\left\| x\right\| = \int_{0}^{1}\left|x(t)\right| d t$ , and $B(x, y) = \int_{0}^{1} x(t) y(t) d t$ . Prove that the functional $B$ is linear and continuous in each variable, but is not continuous on $X \times X$ .

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Problem 7.20

(1)(2)(3)

Let $X$ be a Banach space and $\left\{ A, B, A_{n}, B_{n}: n \geq 1\right\} \subset \mathcal{L}(X)$ . Prove that

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(1)

$A_{n} \xrightarrow {s} A, B_{n} \xrightarrow {s} B \Rightarrow A_{n} B_{n} \xrightarrow {s} A B$ ;

(2)

$A_{n} \rightrightarrows A, B_{n} \rightrightarrows B \Rightarrow A_{n} B_{n} \rightrightarrows A B$ ;

(3)

$A_{n} \xrightarrow {w} A, B_{n} \xrightarrow {w} B \nRightarrow A_{n} B_{n} \xrightarrow {w} A B$ .

Problem 7.21

(1)(2)(3)

Let $X$ be a Banach space, $\left\{ x, x_{n}: n \geq 1\right\} \subset X$ and $\left\{ A, A_{n}: n \geq 1\right\} \subset \mathcal{L}(X)$ . Prove that

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(1)

$A_{n} \xrightarrow {s} A, x_{n} \rightarrow x \Rightarrow A_{n} x_{n} \rightarrow A x$ ;

(2)

$A_{n} \rightrightarrows A, x_{n} \xrightarrow {w} x \Rightarrow A_{n} x_{n} \xrightarrow {w} A x$ ;

(3)

$A_{n} \xrightarrow {s} A, x_{n} \xrightarrow {w} x \nRightarrow A_{n} x_{n} \xrightarrow {w} A x$ .

Problem 7.22

Let $H$ be a Hilbert space, $\left\{ A_{n}: n \geq 1\right\} \subset \mathcal{L}(H), \sup_{n \geq 1}\left\| A_{n}\right\| < +\infty$ , and $\exists L \subset H, \bar{L} = H, \forall x, y \in L: \exists \lim_{n \rightarrow \infty }\left(A_{n} x, y\right)$ .

Prove that there exists $A \in \mathcal{L}(H)$ such that $A_{n} \xrightarrow {w} A$ .

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Problem 7.23

Let $H$ be a separable Hilbert space. Prove that any closed ball in $\mathcal{L}(H)$ is a weakly compact set (i.e., every bounded sequence in $\mathcal{L}(H)$ contains a weakly convergent subsequence).

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Problem 7.24

Let $X$ be a Banach space, $A \in \mathcal{L}(X),\left\{ \lambda_{k}: k \geq 0\right\} \subset \mathbb {R}, \varphi (t) = \sum_{n = 1}^{\infty } \lambda_{n} t^{n}$ , $t \in \mathbb {R}$ , be the sum of a power series which converges on $\mathbb {R}$ . Prove that the sequence $\left\{ \varphi_{n}(A): n \geq 0\right\}$ , where $\varphi_{n}(A) = \sum_{k = 0}^{n} \lambda^{k} A^{k}, n \geq 0, A^{0} = I$ , converges in $\mathcal{L}(X)$ to some operator $\varphi (A) \in \mathcal{L}(X)$ .

Prove that under the condition $\lambda_{k} \geq 0, k \geq 0$ , the inequality $\left\| \varphi (A)\right\| \leq \varphi (\left\| A\right\| )$ holds.

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Problem 7.25

Prove that for a bounded operator $A \in \mathcal{L}(X)$ , where $X$ is a Banach space, the operators

e^{A} = \sum _{k = 0}^{\infty } \frac{A^{k}}{k!}, \sin A = \sum _{k = 0}^{\infty } \frac{(-1)^{k} A^{2 k+1}}{(2 k+1)!}, \cos A = \sum _{k = 0}^{\infty } \frac{(-1)^{k} A^{2 k}}{(2 k)!},

are well-defined, and $e^{A}, \sin A, \cos A \in \mathcal{L}(X)$ .

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Problem 7.26

Let $X$ be a Banach space, $A \in \mathcal{L}(X)$ . Prove that the series $\sum_{n = 0}^{\infty } A^{n}$ converges in $\mathcal{L}(X)$ if and only if for some $k \in \mathbb {N}$ the inequality $\left\| A^{k}\right\| < 1$ holds.

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Problem 7.27

Prove that every linear bounded operator in a Hilbert space is a strong limit of a sequence of bounded finite-dimensional operators if and only if the space is separable.

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Problem 7.28

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

Denote by $T_{n}$ the subspace of the space $\widetilde{C}$ (see problem 7.17*) which consists of trigonometric polynomials of degree at most $n$ . An operator $U \in \mathcal{L}(\tilde{C})$ is called a polynomial (trigonometric) operator of degree $n$ , if $U x \in T_{n}$ for all $x \in \widetilde{C}$ and $U x = x$ for all $x \in T_{n}$ . Prove the following statements:

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(1)

The following operators are polynomial: the operator $S_{n}$ from problem 7.17*; an operator that maps each function to its interpolation (trigonometric) polynomial, constructed for a fixed set of nodes.

(2)

(Zigmund-Marcinkiewicz-Berman) If $U$ is a polynomial operator of degree $n$ , then $\frac{1}{2 \pi } \int_{-\pi }^{\pi }\left(U x^{\tau }\right)(s-\tau ) d \tau = \left(S_{n} x\right)(s), s \in \mathbb {R}, x \in \widetilde{C}$ , where $x^{\tau }(t) = x(t+\tau ), \quad t \in \mathbb {R}, \quad \tau \in \mathbb {R}$ , and for each $s \in \mathbb {R}$ the integrand is a continuous function of $\tau$ .

(3)

If $U$ is a polynomial operator of degree $n$ , then $\left\| U\right\| \geq \left\| S_{n}\right\| > A \ln n, n \geq 1$ , for some constant $A > 0$ .

(4)

The sequence $\left\{ U_{n}: n \geq 1\right\}$ of polynomial operators, where $U_{n}$ is an operator of degree $n$ , does not converge strongly in $\widetilde{C}$ .

Problem 7.29

(1)(2)

A linear operator $A: C(\left[a, b\right]) \rightarrow C(\left[a, b\right])$ , where $C(\left[a, b\right])$ is a real space, is called nonnegative if for all $x \in C(\left[a, b\right])$ such that $x(t) \geq 0$ for all $t \in [a, b]$ , we have $(A x)(t) \geq 0$ for all $t \in [a, b]$ . Prove that

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(1)

The nonnegative operator is continuous, and $\left\| A\right\| = \left\| A e_{0}\right\|$ , where $e_{0}(t) = 1$ , $t \in [a, b]$ .

(2)

(Korovkin) If $\left\{ A_{n}: n \geq 1\right\}$ is a sequence of nonnegative operators, then $A_{n} \xrightarrow {s} I$ in $C(\left[a, b\right])$ if and only if the convergence $A_{n} e_{k} \rightarrow e_{k}, n \rightarrow \infty$ , holds for $e_{k}(t) = t^{k}, t \in [a, b], k = 0,1,2$ .

Is the analogous criterion of weak convergence of nonnegative operators valid?

Problem 7.30

(1)(2)

Let $X = L_{p}(\left[0,1\right])$ or $X = C(\left[0,1\right])$ , the operator $K_{n}: X \rightarrow X$ assigns to each function $x \in X$ the Kantorovich polynomial

\left(K_{n} x\right)(t) = \sum _{k = 0}^{n}\binom {n}{k} x_{n k} t^{k}(1-t)^{n-k}, t \in [0,1]

where $x_{n k} = (n+1) \int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}} x(t) d t, 0 \leq k \leq n, n \geq 1$ . Prove that

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(1)

$K_{n} \xrightarrow {s} I$ in $C(\left[0,1\right])$ ;

(2)

$K_{n} \xrightarrow {s} I$ in $L_{p}(\left[0,1\right]), 1 \leq p < +\infty$ .

Is this statement true for $p = +\infty$ ?