Math Problems Vault

Problem 10.76 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 10.2 "Problems to Solve"

(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 7.4 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 7.2 "Problems to Solve"

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 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 7.2 "Problems to Solve"

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 9.35 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 9.2 "Problems to Solve"

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

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(1)

$A_{n} \rightrightarrows A \Leftrightarrow A_{n}^{*} \rightrightarrows A^{*}$ ;

(2)

$A_{n} \xrightarrow {w} A \Leftrightarrow A_{n}^{*} \xrightarrow {w} A^{*}$ ;

(3)

$A_{n} \xrightarrow {s} A \Leftrightarrow A_{n}^{*} \xrightarrow {s} A^{*}$ .

Problem 9.36 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 9.2 "Problems to Solve"

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

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(1)

if $A_{n} = A_{n}^{*}, n \geq 1, A_{n} \xrightarrow {w} A, n \rightarrow \infty$ , then $A = A^{*}$ ;

(2)

if $A_{n} \geq 0, n \geq 1, A_{n} \xrightarrow {w} A, n \rightarrow \infty$ , then $A \geq 0$ ;

(3)

if $A_{n} \xrightarrow {w} A$ and $\left\| A_{n} x\right\| \rightarrow \left\| A x\right\|$ for all $x \in H$ , then $A_{n} \xrightarrow {s} A$ ; in particular, if $A, A_{n}, n \geq 1$ , are isometric, then $A_{n} \xrightarrow {w} A \Leftrightarrow A_{n} \xrightarrow {s} A$ ;

(4)

if $A_{n}, n \geq 1$ , and $A$ are normal, $A_{n} \xrightarrow {s} A$ , then $A_{n}^{*} \xrightarrow {s} A^{*}$ ;

(5)

if $A_{n}$ are isometric for each $n \geq 1$ , and $A_{n} \xrightarrow {w} A$ , then $A$ is not necessarily isometric;

(6)

if $A_{n}$ are unitary for each $n \geq 1$ , and $A_{n} \xrightarrow {w} A$ , then $A$ is not necessarily unitary.

Problem 9.37 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 9.2 "Problems to Solve"

Let $H$ be a Hilbert space, $\left\{ A_{n}: n \geq 1\right\} \subset \mathcal{L}(H)$ are such that $A_{n} \geq 0, n \geq 1$ , and also there exists $C > 0$ such that $\left\| A_{n}\right\| \leq C, n \geq 1$ . Prove that

A_{n} \xrightarrow {s} 0 \Leftrightarrow A_{n} \xrightarrow {w} 0 \Leftrightarrow \left(A_{n} x, x\right) \rightarrow 0 \text{ for all } x \in H.

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Problem 7.60 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 7.4 "Different types of convergence in the space of operators"

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.

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Problem 7.61 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 7.4 "Different types of convergence in the space of operators"

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.

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Problem 7.62 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 7.4 "Different types of convergence in the space of operators"

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.

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Problem 7.63 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 7.4 "Different types of convergence in the space of operators"

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.

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Problem 7.64 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 7.4 "Different types of convergence in the space of operators"

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.

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Problem 8.56 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 8.2 "Adjoint operators in a Hilbert space. Unitary and normal operators"

Let a sequence $\left\{ A_{n}\right\}_{1}^{\infty }$ of positive operators converge weakly to zero. Prove that $A_{n}$ converge to zero strongly.

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Problem IX.1.1 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section IX.1 "Spectral Measures and Representations of Abelian C*-Algebras"

Let $\mathcal{H}$ be a Hilbert space and let $\{ A_i\}$ be a net in $\mathcal{B}(\mathcal{H})$ . Prove the following statements.

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(a)

$A_i \to A$ (WOT) if and only if $\left\langle A_i h, k \right\rangle \to \left\langle Ah, k \right\rangle$ for all $h, k$ in $\mathcal{H}$ .

(b)

If $\sup_i \left\| A_i\right\| < \infty$ and $\mathcal{T}$ is a total subset of $\mathcal{H}$ , then $A_i \to A$ (WOT) if and only if $\left\langle A_i h, k \right\rangle \to \left\langle Ah, k \right\rangle$ for all $h, k$ in $\mathcal{T}$ .

(c)

$A_i \to A$ (SOT) if and only if $\left\| A_i h - Ah\right\| \to 0$ for all $h$ in $\mathcal{H}$ .

(d)

If $\sup_i \left\| A_i\right\| < \infty$ and $\mathcal{T}$ is a total subset of $\mathcal{H}$ , then $A_i \to A$ (SOT) if and only if $\left\| A_i h - Ah\right\| \to 0$ for all $h$ in $\mathcal{T}$ .

(e)

If $\mathcal{H}$ is separable, then the WOT and SOT are metrizable on bounded subsets of $\mathcal{B}(\mathcal{H})$ .

Problem IX.1.4 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section IX.1 "Spectral Measures and Representations of Abelian C*-Algebras"

If $L: \mathcal{B}(\mathcal{H}) \to \mathbb {C}$ is a linear functional, show that the following statements are equivalent:

$L$ is SOT-continuous;
$L$ is WOT-continuous;
there are vectors $h_1, \dots , h_n, g_1, \dots , g_n$ in $\mathcal{H}$ such that $L(A) = \sum_{j=1}^n \left\langle Ah_j, g_j \right\rangle$ .

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Problem IX.1.5 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section IX.1 "Spectral Measures and Representations of Abelian C*-Algebras"

Show that a convex subset of $\mathcal{B}(\mathcal{H})$ is WOT closed if and only if it is SOT closed.

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Problem IX.1.9 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section IX.1 "Spectral Measures and Representations of Abelian C*-Algebras"

If $\{ E_i\}$ is a net of projections and $E$ is a projection, show that $E_i \to E$ (WOT) if and only if $E_i \to E$ (SOT).

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Problem X.5.1 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section X.5 "Stone's Theorem"

If $U : \mathbb {R} \to \mathcal{B}(\mathcal{H})$ is such that $U(t)$ is unitary for all $t$ , $U(s+t) = U(s)U(t)$ for all $s, t$ , and $U : \mathbb {R} \to (\mathcal{B}(\mathcal{H}), \mathrm{WOT})$ is continuous, then $U$ is SOT-continuous.

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Problem 107 from Halmos P.R. "A Hilbert Space Problem Book" [Hal82]More problems in Chapter 13 "Operator Topologies"

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Remark.

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A Hilbert space has two useful topologies (weak and strong); the space of operators on a Hilbert space has several. The metric topology induced by the norm is one of them; to distinguish it from the others, it is usually called the norm topology or the uniform topology. The next two are natural outgrowths for operators of the strong and weak topologies for vectors. A subbase for the strong operator topology is the collection of all sets of the form

\left\{ A : \left\| (A - A_0)f\right\| < \varepsilon \right\} ;

correspondingly a base is the collection of all sets of the form

\left\{ A : \left\| (A - A_0)f_i\right\| < \varepsilon ,\ i = 1, \ldots , k\right\} .

Here $k$ is a positive integer, $f_1, \ldots , f_k$ are vectors, and $\varepsilon$ is a positive number. A subbase for the weak operator topology is the collection of all sets of the form

\left\{ A : \left| \left\langle (A - A_0)f, g \right\rangle \right| < \varepsilon \right\} ,

where $f$ and $g$ are vectors and $\varepsilon > 0$ ; as above (as always) a base is the collection of all finite intersections of such sets. The corresponding concepts of convergence (for sequences and nets) are easy to describe: $A_n \to A$ strongly if and only if $A_n f \to Af$ strongly for each $f$ (i.e., $\left\| (A_n - A)f\right\| \to 0$ for each $f$ ), and $A_n \to A$ weakly if and only if $A_n f \to Af$ weakly for each $f$ (i.e., $\left\langle A_n f, g \right\rangle \to \left\langle Af, g \right\rangle$ for each $f$ and $g$ ).

The weak topology is smaller (weaker) than the strong topology, and the strong topology is smaller than the norm topology: every weak open set is a strong open set, and every strong open set is norm open. Norm convergence implies strong convergence, and strong convergence implies weak convergence. In the presence of uniformity on the unit sphere, the implications are reversible.

Problem 115 from Halmos P.R. "A Hilbert Space Problem Book" [Hal82]More problems in Chapter 13 "Operator Topologies"

Is every weakly convergent sequence of projections strongly convergent?

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Remark.

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If a net $\left\{ P_n\right\}$ of projections converges weakly to a projection $P$ , then it converges strongly to $P$ . Proof: for each $f$ ,

\left\| P_n f\right\| ^{2} = \left\langle P_n f, f \right\rangle \to \left\langle Pf, f \right\rangle = \left\| Pf\right\| ^{2},

and therefore Problem 20 is applicable.

Problem 491 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection III.5.2 "Operators in Hilbert Space"

Suppose that the sequence $\left\{ A_n\right\}$ of operators on a Hilbert space $H$ converges weakly to $A$ and, moreover, that $\left\| A_nx\right\| \to \left\| Ax\right\|$ for any $x\in H$ . Prove that $\left\{ A_n\right\}$ converges strongly to $A$ . In particular, this means that strong and weak convergence to a unitary limit coincide for sequences of unitary operators.

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Relations between the norm, strong and weak operator topologies

References