Math Problems Vault

Problem 7.1

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

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(1)

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

(2)

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

(3)

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

(4)

$X = C(\left[0,1\right]),\left(A_{n} x\right)(t) = \int_{0}^{1} \sqrt{(t-s)^{2}+\frac{1}{n}} x(s) d s$ ;

(5)

$X = C(\left[0,1\right]),\left(A_{n} x\right)(t) = \int_{0}^{1} t^{n} x(s) d s$ ;

(6)

$X = L_{p}(\mathbb {R}), 1 \leq p < +\infty ,\left(A_{n} x\right)(t) = e^{-(t-n)^{2}} x(t)$ ;

(7)

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

Problem 7.2

Under what conditions on the number $\alpha \in \mathbb {C}$ does the sequence of operators $A_{n}: l_{p} \rightarrow l_{p}, 1 \leq p \leq +\infty$ , defined by the formula

A_{n} x = \left(\alpha ^{n} x_{1}, \alpha ^{n-1} x_{2}, \ldots , \alpha x_{n}, 0,0, \ldots \right), x \in l_{p},

converge uniformly, weakly, strongly?

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Problem 7.3

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

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