Math Problems Vault

Problem 7.59

Let $X$ and $Y$ be normed spaces, sequences of operators $A_{n}$ and $B_{n}$ converging strongly in $\mathcal{B}(X, Y)$ to $A$ and $B$ respectively, and sequences of complex numbers $\alpha_{n}$ and $\beta_{n}$ converging to $\alpha$ and $\beta$ respectively. Prove that $\alpha_{n} A_{n}+\beta_{n} B_{n} \xrightarrow {s} \alpha A+\beta B$ . Prove that if $A_{n} \xrightarrow {s} A$ and $A_{n} \xrightarrow {s} B$ in $\mathcal{B}(X, Y)$ , then $A = B$ . Prove both statements of the problem with strong operator convergence replaced by weak operator convergence.

?

Problem 7.60

Prove that uniform convergence of a sequence of operators from $\mathcal{B}(X, Y)$ implies strong operator convergence, and strong operator convergence implies weak operator convergence.

?

Problem 7.61

Give an example of a sequence of operators of the space $\mathcal{B}(X, Y)$ that converges strongly, but not uniformly. Give an example of a sequence of operators that converges weakly, but not strongly.

?

Problem 7.62

Prove that a sequence of operators $\left\{ A_{n}\right\}_{1}^{\infty }$ from $\mathcal{B}(X, Y)$ converges in norm to an operator $A$ if and only if $A_{n} \xrightarrow {s} A$ uniformly on the unit ball of the space $X$ , i.e., for every $\varepsilon > 0$ there is a number $N$ such that for every $x \in X$ , $\left\| x\right\|_{X} \leqslant 1$ and for all $n > N$ the inequality $\left\| A_{n} x-A x\right\|_{Y} \leqslant \varepsilon$ holds. Prove that a sequence $\left\{ A_{n}\right\}_{1}^{\infty }$ converges in norm to an operator $A$ if and only if $A_{n} \rightharpoonup A$ uniformly on the unit ball of the space $X$ and the unit ball of the space $Y^{*}$ , i.e., for every $\varepsilon > 0$ there is a number $N$ such that for all $x \in X$ , $\left\| x\right\|_{X} \leqslant 1$ and $f \in Y^{*},\left\| f\right\|_{Y^{*}} \leqslant 1$ and for all $n > N$ the inequality $\left|f\left(A_{n} x-A x\right)\right| \leqslant \varepsilon$ holds.

?

Problem 7.63

Let $X$ and $Y$ be normed spaces, with $X$ finite-dimensional. Prove that in the space $\mathcal{B}(X, Y)$ uniform and strong operator convergence coincide.

?

Problem 7.64

Let $X$ and $Y$ be normed spaces, with $Y$ finite-dimensional. Prove that in the space $\mathcal{B}(X, Y)$ strong and weak operator convergence coincide.

?

Problem 7.65

Let $X$ be a normed space, and $Y$ a Banach space. Prove that the space $\mathcal{B}(X, Y)$ is complete with respect to strong operator convergence.

?

Problem 7.66

Let $X$ and $Y$ be normed spaces and let $Y$ be complete with respect to weak convergence. Prove that the space $\mathcal{B}(X, Y)$ is complete with respect to weak operator convergence.

?

Problem 7.67

(H. Hahn, 1922). Let $X$ be a Banach space, and $Y$ a normed space. Let $\mathcal{M} \subset \mathcal{B}(X, Y)$ be a weakly bounded set of operators, i.e., for all $x \in X$ and $f \in Y^{*}$ the inequalities $\sup_{A \in \mathcal{M}}\left|f(A x)\right| < C_{f, x} < \infty$ hold. Prove that $\mathcal{M}$ is bounded in norm. ¹ In particular, any weakly convergent and any weakly fundamental sequence is bounded in norm.

Naturally, this statement remains valid if the condition of weak boundedness is replaced by the condition of strong boundedness. ↩

?

Problem 7.68

Let a sequence of continuous operators $A_{n}$ in a Banach space $X$ converge weakly to $A \in \mathcal{B}(X)$ . Prove that $\left\| A\right\| \leqslant \varlimsup_{n \rightarrow \infty }\left\| A_{n}\right\|$ . Give an example showing that even in the case of strong operator convergence equality may fail.

?

Problem 7.69

Let $X, Y$ and $Z$ be normed spaces, let a sequence $\left\{ A_{n}\right\}_{1}^{\infty } \subset \mathcal{B}(X, Y)$ converge (uniformly, strongly, or weakly) to an operator $A \in \mathcal{B}(X, Y)$ , and let a sequence $\left\{ B_{n}\right\}_{1}^{\infty } \subset \mathcal{B}(Y, Z)$ converge (uniformly, strongly, or weakly) to an operator $B \in \mathcal{B}(Y, Z)$ . In each of the nine possible cases determine whether the sequence $B_{n} A_{n}$ converges to the operator $B A$ , and indicate the type of convergence.

?

Problem 7.70

Prove the criterion for strong convergence of operators. Let $X$ and $Y$ be Banach spaces. A sequence of operators $\left\{ A_{n}\right\}_{1}^{\infty }$ in $\mathcal{B}(X, Y)$ converges strongly to an operator $A$ if and only if two conditions hold:

for every $x \in X$ the sequence $\left\{ A_{n} x\right\}_{1}^{\infty }$ is bounded;
there exists an everywhere dense set $M$ in $X$ such that $A_{n} x \rightarrow A x$ for every $x \in M$ .

?

Problem 7.71

Prove the criterion for weak convergence of operators. Let $X$ and $Y$ be Banach spaces. A sequence of operators $\left\{ A_{n}\right\}_{1}^{\infty }$ in $\mathcal{B}(X, Y)$ converges weakly to an operator $A$ if and only if two conditions hold:

for all $x \in X$ and $f \in Y^{*}$ the sequence $\left\{ f\left(A_{n} x\right)\right\}_{1}^{\infty }$ is bounded;
there exists an everywhere dense set $M$ in $X$ , and an everywhere dense set $L$ in $Y^{*}$ , such that $f\left(A_{n} x\right) \rightarrow f(A x)$ for all $x \in M$ and $f \in L$ .

?

Problem 7.72

Let $X$ and $Y$ be Banach spaces, let the spaces $X$ and $Y^{*}$ (and hence also the space $Y$ ) be separable, and let $Y$ be reflexive. Prove that weak operator convergence on any bounded set of the space $\mathcal{B}(X, Y)$ is metrizable (compare with problems 7.31 and 7.46).

?

Problem 7.73

Let $X$ and $Y$ be Banach spaces, let the spaces $X$ and $Y^{*}$ (and hence also the space $Y$ ) be separable, and let $Y$ be reflexive. Prove that from any bounded sequence of operators from $\mathcal{B}(X, Y)$ one can extract a weakly convergent subsequence (i.e., any bounded set in $\mathcal{B}(X, Y)$ is sequentially precompact with respect to weak operator convergence; compare with problems 7.34 and 7.48).

?

Problem 7.74

Give an example showing that the statement of the previous problem is false for the case of strong operator convergence.

?

Problem 7.75

For the following sequences of operators $A_{n}$ , determine the type of convergence (uniform, strong, weak, or none) and find the limit in the indicated spaces:

?

(a)

$A_{n} \in \mathcal{B}\left(l_{p}\right), p \in [1, \infty ), A_{n}\left(x_{1}, x_{2}, \ldots \right) = \left(0, \ldots , 0, x_{n}, 0,0 \ldots \right)$ ;

(b)

$A_{n} \in \mathcal{B}(X)$ , where $X$ is a normed space, $A_{n} x = \frac{x}{n}$ ;

(c)

$A_{n} \in \mathcal{B}\left(l_{p}\right), p \in [1, \infty ), A_{n}\left(x_{1}, x_{2}, \ldots \right) = \left(0, \ldots , 0, x_{n}, x_{n+1}, \ldots \right)$ ;

(d)

$A_{n} \in \mathcal{B}\left(l_{p}\right), p \in [1, \infty ), A_{n}\left(x_{1}, x_{2}, \ldots \right) = \left(\lambda_{1, n} x_{1}, \lambda_{2, n} x_{2}, \ldots \right)$ , where the vectors $\lambda_{n} = \left(\lambda_{1, n}, \lambda_{2, n}, \ldots \right)$ lie in $l_{\infty }$ and converge in the norm of this space to the vector $\lambda = \left(\lambda_{1}, \lambda_{2}, \ldots \right)$ ;

(e)

$A_{n} \in \mathcal{B}\left(l_{p}\right), p \in [1, \infty ), A_{n}\left(x_{1}, x_{2}, \ldots \right) = \left(\lambda_{1, n} x_{1}, \lambda_{2, n} x_{2}, \ldots \right)$ , where $\left|\lambda_{k, n}\right| < C$ and $\lambda_{k, n} \rightarrow \lambda_{k}$ as $n \rightarrow \infty$ for every $k \in \mathbb {N}$ (i.e., the vectors $\lambda_{n} = \left(\lambda_{1, n}, \lambda_{2, n}, \ldots \right)$ converge to $\lambda = \left(\lambda_{1}, \lambda_{2}, \ldots \right) *$ -weakly in $l_{\infty }$ );

(f)

$A_{n} \in \mathcal{B}(X)$ , where $X = l_{p}, p \in [1, \infty ), X = c_{0}$ or $X = c, A_{n}\left(x_{1}, x_{2}, \ldots \right) = \left(x_{n}, x_{n+1}, \ldots \right)$ (i.e., $A_{n} = T_{l}^{n}$ , where $T_{l}$ is the left shift operator);

(g)

$A_{n} \in \mathcal{B}(C[0,1]),\left(A_{n} x\right)(t) = \int_{0}^{t} \sum_{k = 0}^{n} \frac{s^{k}}{k!} x(s) d s$ ;

(h)

$A_{n} \in \mathcal{B}(C[0,1]),\left(A_{n} x\right)(t) = \int_{1 / n}^{1-1 / n} K(t, s) x(s) d s$ , where $K(t, s)$ is a function continuous on the square $[0,1]^{2}$ ;

(i)

$A_{n} \in \mathcal{B}(C[0,1]),\left(A_{n} x\right)(t) = \int_{0}^{1} K_{n}(t, s) x(s) d s$ , where $K_{n}(t, s)$ are functions continuous on the square $[0,1]^{2}$ , converging uniformly to a function $K(t, s)$ ;

(j)

$A_{n} \in \mathcal{B}(C[0,1]),\left(A_{n} x\right)(t) = \int_{0}^{1} K_{n}(t, s) x(s) d s$ , where $K_{n}(t, s)$ are functions, uniformly bounded in $n$ , continuous on the square $[0,1]^{2}$ , converging pointwise to a continuous function $K(t, s)$ ;

(k)

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

(l)

$A_{n} \in \mathcal{B}(C[0,1]), A_{n} = A^{n}$ , where $(A x)(t) = \int_{0}^{t} K(t, s) x(s) d s$ , and $K(t, s)$ 一 a function continuous on the triangle $\left\{ (t, s): 0 \leqslant s \leqslant t \leqslant 1\right\}$ ;

(m)

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

(n)

$A_{n} \in \mathcal{B}\left(L_{2}[0,1]\right),\left(A_{n} x\right)(t) = \int_{0}^{1} K_{n}(t, s) x(s) d s$ , where $K_{n} \in L_{2}\left([0,1]^{2}\right)$ and $K_{n} \rightarrow K$ in $L_{2}\left([0,1]^{2}\right)$ ;

(o)

$A_{n} \in \mathcal{B}\left(C^{1}[0,1], C[0,1]\right),\left(A_{n} x\right)(t) = \sum_{k = 0}^{n} x(k / n) P_{k, n}(t)$ is the Lagrange interpolation polynomial for the function $x(t)$ , where $P_{k, n}(t) = \prod_{j = 0, j \neq k}^{n} \frac{\left(t-t_{j}\right)}{\left(t_{k}-t_{j}\right)}$ , and $t_{k} = k / n$ ;

(p)

$A_{n} \in \mathcal{B}\left(C_{\text{per }}^{1}[-\pi , \pi ], C[-\pi , \pi ]\right)$ , where $C_{\text{per }}^{1}[-\pi , \pi ]$ is the subspace of the space $C^{1}[-\pi , \pi ]$ consisting of functions satisfying the conditions $x(-\pi ) = x(\pi ), x^{\prime }(-\pi ) = x^{\prime }(\pi ),\left(A_{n} x\right)(t) = \frac{a_{0}}{2} + \sum_{k = 1}^{n}\left(a_{k} \cos k t+b_{k} \sin k t\right)$ is the partial sum of the Fourier series for the function $x(t)$ , where $a_{k} = \frac{1}{\sqrt{\pi }} \int_{-\pi }^{\pi } x(t) \cos k t d t, b_{k} = \frac{1}{\sqrt{\pi }} \int_{-\pi }^{\pi } x(t) \sin k t d t$ ;

(q)

$A_{n} \in \mathcal{B}(C[0,1]),\left(A_{n} x\right)(t) = x\left(t^{1+1 / n}\right)$ ;

(r)

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

\left(A_{n} x\right)(t) = \begin{cases} x(t), & \text{ if } t < 1-1 / n, \\ 0, & \text{ if } t > 1-1 / n ; \end{cases}

(s)

$A_{n} \in \mathcal{B}\left(L_{p}[0,1]\right), p \in [1, \infty ),\left(A_{n} x\right)(t) = \sum_{k = 1}^{n} n \int_{t_{k-1}}^{t_{k}} x(s) d s \chi_{k, n}(t)$ , where $\chi_{k, n}(t)$ is the characteristic function of the segment $\left[t_{k-1}, t_{k}\right]$ , and $t_{k} = k / n$ ;

(t)

$A_{n} \in \mathcal{B}\left(L_{p}[0,1]\right), p \in [1, \infty ],\left(A_{n} x\right)(t) = a_{n}(t) x(t)$ , where the functions $a_{n} \in L_{\infty }[0,1]$ and $a_{n} \rightarrow a$ in $L_{\infty }[0,1]$ ;

(u)

$A_{n} \in \mathcal{B}\left(L_{p}[0,1]\right), p \in [1, \infty ],\left(A_{n} x\right)(t) = a_{n}(t) x(t)$ , where the functions $a_{n} \in C[0,1]$ are uniformly bounded in $n$ in this space and converge pointwise to a function $a(t)$ ;

(v)

\begin{aligned} * A_{n} \in \mathcal{B}\left(L_{p}[-\pi , \pi ]\right) & , p \in [1, \infty ], \\ & \left(A_{n} x\right)(t) = \frac{a_{0}}{2}+\sum _{k = 1}^{n}\left(a_{k} \cos k t+b_{k} \sin k t\right) \end{aligned}

is a partial sum of the Fourier series of the function $x(t)$ , where

a_{k} = \frac{1}{\sqrt{\pi }} \int _{-\pi }^{\pi } x(t) \cos k t d t, \quad b_{k} = \frac{1}{\sqrt{\pi }} \int _{-\pi }^{\pi } x(t) \sin k t d t

(w)

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

\left(A_{n} x\right)(t) = \begin{cases} x(t+1 / n), & \text{ if } t+1 / n \leqslant 1, \\ x(t+1 / n-1), & \text{ if } t+1 / n > 1 ; \end{cases}

(x)

$A_{n} \in \mathcal{B}\left(L_{p}[0,1]\right), p \in [1, \infty ], A_{n} = A^{n},(A x)(t) = x(\varphi (t))$ , where $\varphi (t) \in L_{\infty }[0,1]$ maps the segment $[0,1]$ into itself;

(y)

$A_{n} \in \mathcal{B}\left(L_{p}(\mathbb {R})\right), p \in [1, \infty ],\left(A_{n} x\right)(t) = x(t+1 / n) ;$

(z)

$A_{n} \in \mathcal{B}\left(L_{p}(\mathbb {R})\right), p \in [1, \infty ],\left(A_{n} x\right)(t) = x(t+n)$ .

Problem 7.76

Let $X$ and $Y$ be normed spaces, let a sequence $\left\{ A_{n}\right\}_{n = 1}^{\infty } \subset \mathcal{B}(X, Y)$ converge uniformly, strongly, or weakly to an operator $A \in \mathcal{B}(X, Y)$ , and let a sequence $\left\{ x_{n}\right\}_{n = 1}^{\infty } \subset X$ converge in norm or weakly to a vector $x \in X$ . In each of the six possible cases determine whether the sequence $\left\{ A_{n} x_{n}\right\}_{1}^{\infty }$ converges to the vector $A x$ , and indicate the type of convergence.

?

Problem 7.77

Let $X$ and $Y$ be Banach spaces, $A: X \rightarrow Y$ a linear operator that takes every sequence converging in the norm of $X$ into a weakly convergent sequence from $Y$ . Prove that $A \in \mathcal{B}(X, Y)$ .

?

Problem 7.78

(P. P. Korovkin, 1959). Let $L_{n} \in \mathcal{B}(C[0,1]), n \in \mathbb {N}$ , be positive operators (i.e., for any nonnegative function $x \in C[0,1]$ its images $L_{n} x$ are also nonnegative functions). Prove that if $L_{n}(1) \rightarrow 1, L_{n}(t) \rightarrow t$ , and $L_{n}\left(t^{2}\right) \rightarrow t^{2}$ , then $L_{n} x \rightarrow x$ for any function $x \in C[0,1]$ (i.e., $L_{n} \xrightarrow {s} I$ ).

?

Footnotes